Spin Wavelets on the Sphere

arXiv:0811.2935v3 [math.CA] 15 Dec 2008

Spin Wavelets on the Sphere Daryl Geller Department of Mathematics, University of Stony Brook [email protected]

Domenico Marinucci Department of Mathematics, University of Rome Tor Vergata [email protected]

December 15, 2008 Abstract In recent years, a rapidly growing literature has focussed on the construction of wavelet systems to analyze functions defined on the sphere. Our purpose in this paper is to generalize these constructions to situations where sections of line bundles, rather than ordinary scalar-valued functions, are considered. In particular, we propose needlet-type spin wavelets as an extension of the needlet approach recently introduced by [37, 38] and then considered for more general manifolds by [18, 19, 20]. We discuss localization properties in the real and harmonic domains, and investigate stochastic properties for the analysis of spin random fields. Our results are strongly motivated by cosmological applications, in particular in connection to the analysis of Cosmic Microwave Background polarization data. Keywords and phrases: Wavelets, Frames, Line Bundles, Sphere, Spherical Harmonics, Spin, Needlets, Spin Needlets, High Frequency Asymptotics, Cosmic Microwave Background radiation, Polarization AMS Classification: 42C40, 60G60, 33C55, 14C21, 83F05, 58J05

1

Introduction

In recent years a rapidly growing literature has focussed on the construction of wavelet systems on the sphere, see for instance [3], [51] [1, 2] and the references therein. These attempts have been motivated by strong interest from the applied sciences, for instance in the areas of Geophysics, Medical Imaging and especially Cosmology/Astrophysics. As far as the latter are concerned, special emphasis has been devoted to wavelet techniques for the statistical study of the Cosmic Microwave Background (CMB) radiation data. These data can be viewed as providing observations on the Universe in the immediate sequel to the Big Bang, and as such they have been the object of immense theoretical and applied interest over the last decade [12]. In particular, experiments such as the WMAP satellite from NASA have provided high resolution observations on the ”temperature” (i.e. intensity) of CMB radiation from all directions of the full-sky ([27]). These observations have allowed precise estimates of several parameters of the greatest interest for Cosmology and Theoretical Physics. Spherical wavelets have found very extensive applications here, see for instance ([23, 9, 46, 8, 35, 36, 52]) and many others. The rationale for such a widespread interest can be explained as follows: CMB models are best analyzed in the frequency domain, where the behavior at different multipoles can be investigated separately; on the other hand, partial sky coverage and other missing observations make the evaluation of exact spherical harmonic transforms troublesome. The combination of these two features makes the time-frequency localization properties of wavelets most valuable. Besides providing measurements on the radiation intensity, experiments such as WMAP have also provided some preliminary observations of a much more elusive physical entity, the so-called polarization of the background radiation. Polarization is a property of electromagnetic radiation whose physical significance is described for instance in ([28, 53]), see below for more mathematical discussion. So far,

2 empirical analysis of polarization has been somewhat limited, because this signal is currently measured with great difficulty. The situation with respect to CMB polarization data, however, will significantly improve over the next years, for instance by means of the ESA satellite mission Planck (expected to be launched in Spring 2009) which will take full-sky measurements of the polarized CMB sky ([31]) with much greater precision. Moreover, both ESA and NASA are planning high sensitivity full-sky satellite borne experiments within the next 10-20 years. Polarization measurements are of extreme interest to physicists for several reasons. Indeed, not only do they allow improved precision for estimates of physical parameters which are already the focus of CMB temperature data, but they also open entirely new areas of research. Just to mention a striking example, at large scales the polarization signal is expected to be dominated by a component related to a background of gravitational waves which originated in the Big Bang dynamics (in the so-called inflationary scenario, see ([12]). Detection of this signal would be an outstanding empirical validation of many Theoretical Cosmology claims, directly related to Big Bang dynamic models but deeply rooted in General Relativity. Taking into account the huge amount of polarization data which will be available in the next 1-2 decades, as well as the important cosmological information contained in these data, it is clear that suitable mathematical tools for data analysis are in high demand. While a large amount of mathematical statistics techniques have been developed for analyzing CMB temperature data, far fewer mathematical tools are available to analyze polarization data. From the mathematical point of view, as we shall detail below, polarization can be viewed as a (random) section of a line bundle on the sphere. Our purpose here is then to extend spherical wavelet constructions to the case where sections of line bundles, rather than ordinary functions, are considered. To the best of our knowledge, this is the first attempt to introduce wavelet techniques for the case where one deals with more general mathematical structures than ordinary (scalar-valued) functions on manifolds. We believe this area can be expanded into disciplines other than cosmology - for instance, tensor-valued random fields emerge naturally in brain imaging data ([43]). In particular, in this paper we shall extend to the line bundle case the so-called needlet approach to spherical wavelets. Spherical needlets were recently introduced by [37, 38], and further developed, and extended to general smooth compact Riemannian manifolds in [18, 19, 20]. In a random fields environment, needlets were investigated by ([4, 5]), with a view to applications to the statistical analysis of CMB data; applications in the physical literature include [41], [33], [25], see also ([29], [11, 14, 15]) and [30, 34]. More precisely, we will focus on needlet-type spin wavelets, and we will argue below that they enjoy properties analogous to those of the usual needlets in the standard scalar case. In particular, we shall show below that needlet-type spin wavelets enjoy both the localization and the uncorrelation properties that make scalar needlets a powerful tool for the analysis of scalar-valued spherical random fields. More details on the plan and significance of our paper are given in the next section. The companion article [17], intended primarily for physicists, goes into further detail about the statistical uses of spin wavelets. One should be aware, however, that there are some notational differences between that article and this one.

2

Plan of the Paper and Significance of the Results

The spin s concept that we will use in this paper originates in the fundamental work of Newman and Penrose [39]. Writing for physicists, they say that a quantity η defined on the sphere has spin weight s ∈ Z, provided that, whenever a tangent vector m at a point transforms under coordinate change by m′ = eiψ m, the quantity η, at that point, transforms by η ′ = eisψ η. They then develop a theory of spin s spherical harmonics, upon which we shall build to produce spin wavelets. In section 3, we put the notion of spin s into acceptable mathematical language. We do this in an elementary manner which remains very close to the spirit of the work of Newman and Penrose. For another approach, using much more machinery, see [13]. Let N be the north pole (0, 0, 1), let S be the south pole, (0, 0, −1), and let UI be S 2 \ {N, S}. If

3 R ∈ SO(3), we define UR = RUI . On UI we use standard spherical coordinates (θ, φ) (0 < θ < π, −π ≤ φ < π), and analogously, on any UR we use coordinates (θ R , φR ) obtained by rotation of the coordinate system on UI . At each point p of UR we let ρR (p) be the unit tangent vector at p which is tangent to the circle θR = constant, pointing in the direction of increasing φR . (Thus, for any p ∈ UI , ρR (Rp) = R∗p [ρI (p)].) If p ∈ UR1 ∩ UR2 , we let ψ pR2 R1 be the angle from ρR1 (p) to ρR2 (p). Now say Ω ⊆ S 2 is open. We say that f = (fR )R∈SO(3) ∈ Cs∞ (Ω), or that f is a smooth spin s function on Ω, provided all fR ∈ C ∞ (UR ∩ Ω), and that for all R1 , R2 ∈ SO(3) and all p ∈ UR1 ∩ UR2 ∩ Ω, fR2 (p) = eisψ fR1 (p),

(1)

where ψ = ψ pR2 R1 . If, say, R1 = I, R2 = R, then heuristically fR is fI “looked at after the coordinates have been rotated by R”; at p, it has been multiplied by eisψ . The angle ψ would be clearly be the same if we had instead chosen ρR (p) to point in the direction of increasing θR , say, so this is an appropriate way to make Newman and Penrose’s concept precise. Note that if s = 0, we can clearly identify Cs∞ (Ω) with C ∞ (Ω). For any s, we can identify Cs∞ (Ω) with the sections over Ω of the complex line bundle obtained by using the eisψ pR2 R1 as transition functions from the chart UR1 to the chart UR2 ; then fR is the trivialization of the section over UR ∩ Ω. This is the point of view that we take in section 3, since, as is often the case in mathematics, certain properties are clearer, and more easily verified, if one uses the coordinate-free line bundle point of view. However, right now, for the reader’s convenience, we state our results without reference to line bundles. Line bundles are rarely mentioned explicitly in this article after section 3. We may define L2s (Ω) (resp. Cs (Ω)), by requiring that the fR be in L2 (UR ∩ Ω) (resp. C(UR ∩ Ω)) instead of C ∞ (UR ∩ Ω). There is a well-defined inner product on L2s (Ω), given by hf, gi = hfR , gR i; clearly this definition is independent of choice of R. A key observation of section 3 is that there is a unitary action of SO(3) on L2s (S 2 ), given by f → f R , which is determined by the equation (f R )I (p) = fR (Rp). We think of f R as a “rotate” of f . In section 4, we explain the spin s theory of Newman and Penrose, within the rigorous framework we have just outlined. Most of our arguments are very close to theirs. For f smooth as above, following Newman and Penrose we define i ∂ ∂ (sin θR )−s fR , + ∂sR fR = −(sin θR )s ∂θR sin θR ∂φR ∞ and we show that the “spin-raising” operator ∂ : Cs∞ (Ω) → Cs+1 (Ω) given by (∂f )R = ∂sR fR is well-

defined. We also define ∂ sR fR by ∂ sR fR = ∂sR fR , which leads to the spin-lowering operator ∂ : Cs∞ (Ω) → ∞ Cs−1 (Ω) given by (∂f )R = ∂ sR fR . We show that ∂ and ∂ commute with the actions of SO(3) on smooth spin functions. Now, for l ≥ |s|, let bls = [(l + s)!/(l − s)!]1/2 for s ≥ 0, bls = [(l − s)!/(l + s)!]1/2 for s < 0. Let {Ylm : l ≥ 0, −l ≤ m ≤ l} be the standard basis of spherical harmonics on S 2 . Following Newman and Penrose, for l ≥ |s|, we define the spin s spherical harmonics by s Ylm = ∂ s Ylm /bls for s ≥ 0, s s Ylm = (−∂) Ylm /bls for s < 0, and we show that {s Ylm : l ≥ |s|, −l ≤ m ≤ l} forms an orthonormal 2 basis for Ls (S 2 ). There is a (relatively) simple explicit expression for the s YlmI ; it has the form s YlmI (θ, φ)

=

s ylm (θ)e

imφ

(2)

for a suitable function s ylm . (See (60) for the explicit formula.) If R 6= I, s YlmR does not have a simple R R expression; but s YlmR (θ R , φR ) equals s ylm (θR )eimφR in the coordinates on UR . (Here again s Ylm denotes the “rotate” of s Ylm .) There is also an analogue ∆s of the spherical Laplacian for spin s functions; specifically, if s ≥ 0, we let ∆s = −∂∂, while if s < 0, we let ∆s = −∂∂. Then ∆0 is the usual spherical Laplacian. For l ≥ |s|, let Hls =< s Ylm : −l ≤ m ≤ l >. Also, for l ≥ |s|, let λls = (l − s)(l + s + 1) if s ≥ 0, and let λls = (l + s)(l − s + 1) if s < 0. Then Hls is the subspace of Cs∞ (S 2 ) which consists of eigenfunctions of

4 ∆s for the eigenvalue λls . This of course generalizes the situation in which s = 0. All of the aforementioned facts follow by making the arguments of Newman and Penrose rigorous. In sections 5, 6, 7 and 8, we present new results. In section 5, we define operators with smooth kernels, from Cs∞ (S 2 ) to itself. We say K = (KR′ ,R )R′ ,R∈SO(3) is such a smooth kernel if each ′ KR′ ,R ∈ C ∞ (S 2 ×S 2 ), and if KR1 ′ ,R1 (x, y) = eis(ψ −ψ) KR′ ,R (x, y) for all x ∈ UR1 ′ ∩UR′ and y ∈ UR1 ∩UR , where ψ ′ = ψ xR′1 R′ and ψ = ψ yRR1 . It is then evident that we may consistently define an operator R K : Cs∞ (S 2 ) → Cs∞ (S 2 ) by (Kf )R′ (x) = S 2 KR′ ,R (x, y)fR (y)dS(y), for all x ∈ UR′ ; we call K the operator with kernel K. (These operators may be identified with operators with smooth kernels acting on sections of the aforementioned line bundle.) Next we show that there is a good notion of spin s zonal harmonic. To see how this arises, we first show (in section 3) that, if f ∈ Cs (S 2 ), then the function eisφ fI (θ, φ) extends continuously from UI to UI ∪ N, and the function e−isφ fI (θ, φ) extends continuously from UI to UI ∪ S. (This is shown by considering fR for those R with N, S ∈ UR .) Thus there is a well defined linear functional L : Cs (S 2 ) → C, given by Lf = limθ→0+ eisφ fI (θ, φ). It is evident from (2) that L(s Ylm ) must be zero if m 6= −s. This indicates that s Yl,−s could have a special status; in fact, it is a multiple of the spin s zonal harmonic, in the sense of the following results, which we prove in section 5. +p (2l + 1)/4π s Yl,−s . Then: Lemmas on Zonal Harmonics Define s Zl = (−1)s (a) L(s Zl ) = (2l + 1)/4π. (b) For any f ∈ Hls , Lf = hf, s Zl i. Next, let Pls be the projection in L2 (S 2 ) onto Hls , so that Pls has kernel K ls , where for any rotations Pl ls R, R′ , KR ′ ,R (x, y) = m=−l s YlmR′ (x)s YlmR (y) for x ∈ UR′ , y ∈ UR . Then: ls (c) If x ∈ UI ∩ UR′ and N ∈ UR , then s ZlI (x) = eis(ψ1 −ψ2 ) KR ′ ,R (x, N), where ψ 1 = ψ xR′ I , and ψ 2 is the angle from ρR (N) to ∂/∂y at N. ls (d) If x ∈ UR , then KR,R (x, x) = (2l + 1)/4π.

In section 6, we begin our study of spin wavelets. Before explaining our construction, we need to explain the ideas which are used in the spin 0 case on the sphere, and on more general manifolds. The references here are [37], [38], [18], [19], [20]. A word about notation: in sections 3-5 we often use the variable “f ” to denote a spin function on the sphere. In sections 6-8 the variable “f ” will be reserved for another purpose. Specifically, say f ∈ S(R+ ), f 6= 0, and f (0) = 0. Let (M, g) be a smooth compact oriented Riemannian manifold, and let ∆ be the Laplace-Beltrami operator on M (for instance, the spherical Laplacian if M is S 2 ). Let Kt be the kernel of f (t2 ∆). Then, as we shall explain, the functions wt,x (y) = K t (x, y),

(3)

if multiplied by appropriate weights, can be used as wavelets on M. In case M = S n and f has compact support away from the origin, we shall say these wavelets are “needlet-type”; the theory of needlets was developed in [37], [38]. (Actually, the definition of needlet in [37], [38] is slightly different from (104), as we shall explain.) The theory was worked out in full generality in [18], [19], [20]. R ∞ Specifically, one starts with the Calder´ on formula: if c ∈ (0, ∞) is defined by c = 0 |f (t)|2 dt t , then for all u > 0, Z ∞ dt (4) |f (tu)|2 = c < ∞. t 0 Discretizing (4), if a > 1 is sufficiently close to 1, one obtains a special form of Daubechies’ condition: for all u > 0, ∞ X |f (a2j u)|2 ≤ Ba < ∞, (5) 0 < Aa ≤ j=−∞

5 where c 1 − O(|(a − 1)2 (log |a − 1|)| , 2| log a| c 1 + O(|(a − 1)2 (log |a − 1|)|) . Ba = 2| log a| Aa =

(6) (7)

((6) and (7) were proved in [16], Lemma 7.6. In particular, Ba /Aa converges nearly quadratically to 1 as a → 1. For example, Daubechies calculated that if f (u) = ue−u and a = 21/3 , then Ba /Aa = 1.0000 to four significant digits. Let P be the projection in L2 (S 2 ) onto the space of constant functions (the null space of ∆.) Suppose that a > 1 is sufficiently close to 1 that Aa and Ba are close (or that a, f have been chosen in such a manner that Ba = Aa ). Then for an appropriate discrete set {xj,k }(j,k)∈Z×[1,Nj ] on M, and certain weights µj,k , the collection of {φj,k := µj,k waj ,xj,k } constitutes a wavelet frame for (I − P )L2 (S 2 ). By this we mean that there exist “frame bounds’ 0 < A ≤ B < ∞, such that for any F ∈ (I − P )L2 (S 2 ) the following holds: X A k F k22 ≤ | hF, φj,k i |2 ≤ B k F k22 . (8) j,k

If the points {xj,k } are selected properly, B can be made arbitrarily close to Ba and A can be made arbitrarily close to Aa . Because of the proximity of Aa to Ba , the frame is therefore a “nearly tight frame”. In passing from one scale to another, we use “dilations” in the sense that we are looking at the kernel of f (t2 ∆) for different t, which corresponds to dilating the metric on the manifold. On the sphere, the wavelets at each scale j are all rotates of each other, up to constant multiples. (This is analogous to the usual situation in which all the wavelets at each scale are translates of each other.) Again, wt,x is given by (3). On the sphere S 2 , we have Kt (x, y) =

∞ X l X

f (t2 λl )Ylm (x)Ylm (y).

(9)

l=0 m=−l

The most important cases to consider on the sphere are the case in which f has compact support away from 0 (the “needlet-type” case, of Narcowich, Petrushev and Ward), and the case in which f (u) = ur e−u for some integer r ≥ 1 (the “Mexican needlet” case, as considered in [18], [19]). (Actually, in their work, Narcowich, Petrushev and Ward used l2 in place of l(l + 1) in (105), but this is a minor distinction.) Needlet-type wavelets and Mexican needlets each have their own advantages. Needlet-type wavelets have these advantages: for appropriate f , a, µj,k and xj,k , needlet-type wavelets are a tight frame on the sphere (i.e., A = B in (8)), and the frame elements at non-adjacent scales are orthogonal. In fact, for appropriate needlet-type f , a, µj,k and xj,k , one can arrange that Aa = Ba = A = B = 1 in the discussion above. In that case, the wavelets are called needlets. Mexican needlets have their own advantages. In [18], [19], an approximate formula is written down for them which can be used directly on the sphere. (This formula, which arises from computation of a Maclaurin series, has been checked numerically. It is work in progress, expected to be completed soon, to estimate the remainder terms in this Maclaurin series. In [18], [19] the formula is written down only for r = 1, but it can be readily generalized to general r. The formula indicates that Mexican needlets are quite analogous to the Mexican hat wavelet which is commonly used on the real line.) Mexican needlets have Gaussian decay at each scale, and they do not oscillate (for small r). Thus they can be implemented directly on the sphere, which is desirable if there is missing data (such as the “sky cut” of the CMB). It would be worthwile to utilize both needlets and Mexican needlets in the analysis of CMB, and the results should be compared. In this article we focus on needlet-type situations. A key property of this kind of wavelet is its localization. Working in the general situation (general M, general f ), one has:

6 for every pair of C ∞ differential operators X (in x) and Y (in y) on M, and for every nonnegative integer N , there exists c := cN,X,Y such that for all t > 0 and x, y ∈ M |XY Kt (x, y)| ≤ c

t−(n+I+J) , (d(x, y)/t)N

(10)

where I := deg X, J := deg Y , and d(x, y) denotes the geodesic distance from x to y. In the case where M is the sphere, f has compact support away from 0, and if one replaces λl = l(l + 1) by l2 in the formula for Kt , this was shown by Narcowich, Petrushev and Ward, in [37] and [38]. The general result was shown in [18]. The proof of (10) in [18] would in fact go through without change if, instead of Kt being the kernel of f (t2 ∆), where ∆ is the Laplace-Beltrami operator, one assumed that Kt was the kernel of f (t2 D), where D is any smooth positive second-order elliptic differential operator, acting on C ∞ (M). It is natural to conjecture then that something analogous holds if D is instead a smooth positive second-order elliptic differential operator, between sections of line bundles on M. Since ∆s may be interpreted as being such an operator, it is natural to want to prove the following result. Theorem 6.1 Let Kt be the kernel of f (t2 ∆s ). Then: For every R, R′ ∈ SO(3), every pair of compact sets FR ⊆ UR and FR′ ⊆ UR′ , and every pair of C ∞ differential operators X (in x) on UR′ and Y (in y) on UR , and for every nonnegative integer N , there exists c such that for all t > 0, all x ∈ FR′ and all y ∈ FR , we have |XY Kt,R′ ,R (x, y)| ≤ c

t−(n+I+J) , (d(x, y)/t)N

(11)

where I := deg X and J := deg Y . In this article, we only give the proof of Theorem 6.1 in the easier “needlet-type” case, where f has compact support away from the origin. The general case will be established elsewhere. Let us now explain what we mean by spin wavelets. For R ∈ SO(3), and x ∈ UR , let us define XX wtxR = f (t2 λls ) s YlmR (x) s Ylm .

(12)

l≥|s| m

Then wtxR ∈ Cs∞ (S 2 ). Also note that wtxR,R′ (y) = K t,R,R′ (x, y) (if x ∈ UR and y ∈ UR′ ); this generalizes the case s = 0 of (3). Moreover, if F ∈ L2s (S 2 ), then for x ∈ UR β t,F,x,R := hF, wtxR i = (f (t2 ∆s )F )R (x) = (β t,F )R (x) where we have set β t,F = (f (t2 ∆s )F ). P P 2 2 l≥|s| m alm s Ylm ∈ Ls (S ), then

β t,F,x,R =

We call β t,F,x,R a spin wavelet coefficient of F . XX

f (t2 λls )alm s YlmR (x).

(13) If F =

(14)

l≥|s| m

Let us now explain how, in analogy to the case s = 0, one can obtain a nearly tight frame from spin wavelets. Let P = P|s|,s be the projection onto the H|s|,s , the null space of ∆ls (in Cs∞ ). Suppose that a > 1 is sufficiently close to 1 that Aa and Ba (of (6) and (7)) are close (or that a, f have been chosen in such a manner that Ba = Aa ). We then claim that for an appropriate discrete set {xj,k }(j,k)∈Z×[1,Nj ] on S 2 , certain Rj,k with xj,k ∈ URj,k , and certain weights µj,k , the collection of {φj,k := µj,k waj ,xj,k ,Rj,k } constitutes a nearly tight wavelet frame for (I − P )L2s (S 2 ). (Note that this property is independent

7 of the P choice of Rj,k .) In that case, we will have that for some C, if F ∈ (I − P )L2s , then, in L2 , F ∼ C j,k hF, φj,k iφj,k (note that this sum is independent of the choice of Rj,k ). The precise statement is Theorem 6.3 below, where we explain how the xj,k and µj,k are to be chosen. Theorem 6.3 will be proved elsewhere [21]; in this article, it is only used in section 8. One can certainly choose a, and f with compact support away from the origin, so that Aa = Ba = 1 (recall (5)). Such f were used in [37] and [38]. It is then natural to conjecture, that as in the spin 0 case of [37] and [38], one can choose the points xj,k so that the {µj,k ϕj,k } are an exactly tight frame for (I − P )L2s (S 2 ), for suitable weights µj,k . One should note, however, that even if such explicit points and weights could be found, it might not be practical for physicists to use them in CMB analysis, where measurements are invariably taken at the HEALPix points [24]. This, then, is a situation where it is very useful (and perhaps essential) to have the flexibility in choice of the xj,k which Theorem 6.3 allows. In section 7, we begin to examine how our theory can be applied to cosmology. We look at random spin s fields G on S 2 . We say that G is isotropic (in law) if for every x1 , . . . , xN ∈ UI , the joint probability R distribution of GR I (x1 ), . . . , GI (xN ) is independent of R ∈ SO(3). Similarly, if G, H are both random spin s fields, we say that G, H are jointly isotropic (in law) if for every x1 , . . . , xN , y1 , . . . , yM ∈ UI , the R R R joint probability distribution of GR I (x1 ), . . . , GI (xN ), GI (y1 ), . . . , GI (yM ) is independent of R ∈ SO(3). Say G, H are jointly isotropic, and denote their spin s spherical harmonic coefficients by alm = hGR , s Ylm i, blm = hH R , s Ylm i; these are random variables. In Theorem 7.2 we show that E(alm bl′ m′ ) = 0 unless l = l′ and m = m′ (see [6] for the scalar case). Moreover E(alm blm ) does not depend on m; we will denote it by Cl,G,H . In Theorem 7.3, we generalize a key result in [4], to show that the spin wavelet coefficients of G, H are asymptotically uncorrelated, under mild hypotheses. That is, for x ∈ UR , we define the spin wavelet coefficients β G,t,x,R =< G, wtxR >, β H,t,x,R =< H, wtxR >, and we show that |Cor(β G,t,x,R , β H,t,y,R )| → 0 as t → 0+ . (Again we assume that we are in the “needlet-type” situation, where f has compact support away from 0.) Finally, in section 8, we offer a brief glimpse of the consequences of the considerations of section 7 for stochastic limits and cosmology. Say ǫ > 0. We use a nearly tight frame, as provided by Theorem P 6.3. Specifically, in that theorem, we 2 2j fix a real f supported in [1/a2 , a2 ] for which the Daubechies sum ∞ j=−∞ f (a u) = 1 for all u > 0, so that Aa =P Ba = 1. We then produce the {φj,k } = {φj,k,ǫ } with frame bounds B = 1 + ǫ and A = 1 − ǫ. Say G = Alm s Ylm is an isotropic P random spin s field, and let Cl = Cl,G,G (notation as in Theorem 7.3). For j ∈ Z, we also let γ = j a−2 ≤a2j λls ≤a2 Cl (2l + 1); this is a small quantity since P P C (2l + 1) = 3VarG. (In all sums here, l satisfies the implicit restriction l ≥ |s|.) We let γ ≤ 3 l j l j β jk = β jkǫ = hG, φj,k,ǫ i. We focus on the quadratic statistics X 2 X X ˜j = β jk = β jkǫ 2 , bj = Γ Γ f 2 (a2j λls )|Alm |2 (15) k

k

l,m

˜ j , we can even let ǫ depend on j here. In Proposition 8.1, we show that When considering Γ bj − Γ ˜ j |) ≤ ǫγ j E(|Γ

(16)

for all j. ˜ j is a Since we can let ǫ 0; and (2) sgn ψ = sgn g(Jv, w). Now, say we have an atlas U on M. Suppose that for each chart Uα ∈ U we have a section ρα of T Uα (the real tangent bundle of Uα ). (This section will give, at each point of Uα , a “reference direction”.) Set ρ = {ρα }Uα ∈U . Say also s ∈ Z. We then define Ls , the spin s line bundle associated to (M, U, ρ), as follows. Say Uα , Uβ are charts in U, and define, for each p ∈ Uα ∩ Uβ , ψ pβα to be the angle from ρα (p) to ρβ (p). We then set λβα (p) = eisψ pβα . Then λαβ λβα = 1 on Uα ∩ Uβ , and if Uγ is a chart in U as well, then λαγ λγβ λβα = 1 on Uα ∩ Uβ ∩ Uγ . By a standard argument (see, for instance, the argument on [26], pages 139-140), the λβα may be used as transition functions to define a complex (not necessarily holomorphic) line bundle Ls ; this bundle is unique up to isomorphism. If π is the projection in Ls onto M, there are therefore diffeomorphisms Φα : π −1 (Uα ) → Uα × C, such that if p ∈ Uα ∩ Uβ , and if Φα (p, η) = (p, z),

10 then Φβ (p, η) = (p, eisψ pβα z). Let us say that these maps {Φα } are induced by (U, ρ). Note that, if Ω is a open subset of M, we may naturally restrict Ls to become a line bundle LsΩ over Ω, by using the charts {Uα ∩ Ω : U ∈ U}, and the section ρ′α := ρα on Uα ∩ Ω. Suppose now that Ls1 , Ls2 are determined, respectively, by (M1 , U 1 , ρ1 ) and (M2 , U 2 , ρ2 ). Say F : M1 → M2 is holomorphic, and is a local diffeomorphism. Then F naturally gives rise to a smooth map Fs : Ls1 → Ls2 as follows. Say (U 1 , ρ1 ) induces the maps {Φ1α }, and (U 2 , ρ2 ) induces the maps {Φ2β }. Say p ∈ Uα1 , a chart in U 1 , and q = F (p) ∈ Uβ2 , a chart in U 2 . The map F gives rise to a mapping F∗p : M1p → M2q . Let ψ be the angle from F∗p [ρ1α (p)] to ρ2β (q). Say now (p, η) ∈ π −1 (Uα1 ), and that Φ1α (p, η) = (p, z). Then we define Fs (p, η) = (q, ζ), where Φ2β (q, ζ) = (q, eisψ z). Since F∗ preserves angles, it is easy to see that this definition is independent of choice of α, β. Further, if I : M1 → M1 is the identity map, so is Is : Ls1 → Ls2 . Moreover, say F is actually a biholomorphic map. Then, since F∗ and (F −1 )∗ preserve angles, one easily sees that (Fs )−1 = (F −1 )s . If f is a (continuous) section of Ls2 , we may naturally define a section f F of Ls1 by f F = Fs−1 ◦ f ◦ F. (17) If G : M0 → M1 is another biholomorphic map, we then have (f F )G = f F ◦G .

(18)

This is all quite abstract, but in fact we shall study spin s bundles only when the manifold is an open subset of the sphere S 2 or the complex plane C. On S 2 , by choosing the atlas and ρ well, we can impose a very interesting and very explicit structure. On S 2 , realized as {(x, y, z) ∈ R3 : x2 + y 2 + z 2 = 1}, we let N be the north pole (0, 0, 1), and let S be the south pole, (0, 0, −1). We define the chart UI to be S 2 \ {N, S}. (The choice of chart map from UI to C is irrelevant; one can take it to be a stereographic projection.) We obtain all other charts in our atlas by rotating UI . Thus, if R ∈ SO(3), we define UR = RUI . On UI we often use standard spherical coordinates (θ, φ), and at each point p of UI we let ρI (p) be the unit tangent vector at p which is tangent to the circle θ = constant. (For definiteness, if this circle is x = r cos φ, y = r sin φ, z = constant, let us choose ρI (p) to point in the direction in which φ increases, the “counterclockwise” direction.) On any other chart UR , we choose ρR (Rp) = R∗p [ρI (p)]. (19) for any p ∈ U0 . On the chart UR we can use coordinates (θR , φR ), obtained by rotation of the (θ, φ) coordinates. Thus, if the (θ, φ) coordinates of p ∈ UI are (θ 0 , φ0 ), then the (θ R , φR ) coordinates of Rp ∈ UR are also (θ0 , φ0 ). Then, at any q ∈ UR , ρR (q) is the unit tangent vector to the circle φR = constant, pointing in the direction of increasing φR . Again, the angle ψ pR2 R1 is the angle from ρR1 (p) to ρR2 (p). We would clearly obtain the same angle if ρR (p) had been chosen as the unit vector pointing in the direction of increasing θR , or in the direction of any fixed linear combination of the unit vectors pointing in the directions of increasing φR and increasing θ R . In this precise sense, ψ pR2 R1 effectively measures the angle by which the tangent plane at p is rotated if one uses the (θ R2 , φR2 ) coordinates instead of the (θR1 , φR1 ) coordinates. This is why one should think of our choice of ρR (p) only as a convenient “reference direction”.

11 On C we will use only a single chart, C itself, and if p ∈ C, we let our reference direction ρ(p) be ∂/∂y. Since there is only one chart, this bundle is trivial. However, if F maps an open subset U of C conformally onto another open subset U of C, we can consider the map Fs , which, as we shall see, can be quite interesting. Now, if f is a (continuous) section of Ls over an open set Ω ⊆ S 2 , then we may write ΦR (f (q)) = (q, fR (q)) for all q ∈ UR ∩ Ω, for a suitable function fR : UR ∩ Ω → C. Note, then, that if p ∈ UI ∩ UR ∩ Ω, then fR (p) = eisψ fI (p)

(20)

ψ is the angle from ρI (p) to ρR (p) = R∗p (ρI (R−1 p)).

(21)

where 2

Note also that if Ω ⊆ S , and R0 ∈ SO(3), any smooth function f(R0 ) : UR0 ∩ Ω → C determines a unique section f of Ls over UR0 ∩ Ω with fR0 = f(R0 ) . Indeed, say R ∈ SO(3). If p ∈ UR0 ∩ UR ∩ Ω, then fR (p) must be given by (22) fR (p) = eisψ fR0 (p). where of course, fR0 = f(R0 ) , and where ψ is the angle from ρR0 (p) to ρR (p).

(23)

Conversely, there is surely a section f of Ls over UR0 ∩ Ω with these fR . Consequently the space of smooth sections f of Ls over Ω may be identified with the space {f = (fR )R∈SO(3) : each fR ∈ C ∞ (UR ∩Ω), and (22), (23) hold for all R0 , R ∈ SO(3) and all p ∈ UR0 ∩UR ∩Ω}. (24) We shall frequently make this identification, which often greatly simplifies matters conceptually. However, as is often the case in mathematics, certain properties are clearer, and more easily verified, if one uses the coordinate-free line bundle point of view. Equation (20) describes what happens when one “rotates the charts”, and we now argue that “you get the same answer if you rotate the section instead”. Thus, if R ∈ SO(3), as we have seen, R : S 2 → S 2 gives rise to a smooth map Rs : Ls → Ls . If f is a section of Ls over Ω ⊆ S 2 , using (17) we obtain the “rotated section” f R = Rs−1 ◦ f ◦ R. (25) over R−1 Ω. (Note that if s = 0, f R = f ◦ R.) Thus, for p ∈ Ω,

(f R )(R−1 p) = Rs−1 ◦ f (p), from which we see that (f R )I (R−1 p) = eisψ fI (p)

(26)

−1 ψ is the angle from R∗p (ρI (p)) to ρI (R−1 p).

(27)

where Note that this ψ is the same as the ψ in (21), since R∗p preserves angles. Therefore fR (p) = (f R )I (R−1 p).

(28)

One could take the point of view of (24) here, and simply define f R by setting (f R )I (q) = fR (Rq). But then it would be tedious (though certainly possible) to verify that the smoothness of f implies the smoothness of f R , or that (f R1 )R2 = f R1 R2 , both of which are evident from (25).

12 We also need to discuss the metric and the orientation we will use on S 2 . Since rotations are holomorphic maps on S 2 , any metric in the intrinsic conformal class on S 2 is rotationally invariant, up to a constant multiple at each point. Thus there is a truly rotationally invariant metric in this conformal class, obtained by rotation of the metric at any particular point on the sphere. We note that such a rotationally invariant metric coincides (up to a constant multiple) with the Euclidean metric, as restricted to tangent P P ∂ ∂ vectors on the sphere. (That is, for some positive constant c, if v = 3j=1 aj ∂x and w = 3j=1 bj ∂x j j are real tangent vectors at any point on the sphere, their inner product there is c < v, w > , where E P3 < v, w >E = j=1 aj bj .) Indeed, since < v, w >E is rotationally invariant, this fact need only be checked at the South pole S. We can determine a metric in the intrinsic conformal class on S 2 , by pulling back the standard metric on C through stereographic projection from the North Pole onto the tangent plane to the sphere at S; denote this stereographic projection by T . It is geometrically evident, and it is easily calculated (see the formula (31) below for σ = T /2), that T∗S (∂/∂x) = ∂/∂x, and T∗S (∂/∂y) = ∂/∂y. ∂ ∂ Thus the tangent vectors ∂x and ∂y , are, up to a choice of conformal factor at S, an orthonormal basis at S, as desired. We use < v, w >E as our metric on S 2 . We choose the orientation on S 2 which makes T orientation-preserving. Thus (∂/∂x, ∂/∂y) is an oriented orthonormal basis at S, and, consequently, (∂/∂y, ∂/∂x) is an oriented orthonormal basis at N. In our first new result about the spin s bundle over S 2 , we take a section f of Ls , and examine the behavior of fI as we approach the north and south poles. Theorem 3.1 Let f be a continuous section of Ls over S 2 . Then the function eisφ fI (θ, φ) extends continuously from UI to UI ∪ N, and the function e−isφ fI (θ, φ) extends continuously from UI to UI ∪ S. Proof Let us show the second statement; the proof of the first statement is almost identical. For any φ ∈ [0, 2π), let fI,φ (θ) = fI (θ, φ). It will suffice to show that limθ→π− e−isφ fI,φ (θ) exists, uniformly in φ, and that this limit is independent of φ. For φ ∈ [0, 2π), let γ φ be the great arc γ φ (θ) = (θ, φ) (0 ≤ θ ≤ π). If p = γ φ (θ) is on this great arc, then ρI (p) points in the direction γ ′φ (θ) × p (cross product); thus lim ρI (γ φ (θ)) := vφ

θ→π −

exists, and this limit is uniform in φ (indeed, a rotation about N takes any one of these limiting situations into any other). Moreover, evidently, vφ is a tangent vector at S, and given our choice of orientation at S, the angle from v0 to vφ is φ. (29) Now, let R be any rotation of S 2 which takes S to a point other than N or S. We then have that, for (θ, φ) ∈ UI ∩ UR , fI,φ (θ) = e−isψθ,φ fR (θ, φ) where ψ θ,φ is the angle from ρI (p) to ρR (p), if p = (θ, φ). For fixed φ, the limit as θ → π − of the right side exists (uniformly in φ) and equals e−isΨφ fR (S), where now Ψφ is the angle from vφ to ρR (S).

(30)

Thus for any φ, lim e−isφ fI,φ (θ) = e−is(φ+Ψφ) fR (S) = e−is(φ+Ψφ−Ψ0 ) e−isΨ0 fR (S) = e−is(φ+Ψφ−Ψ0 ) lim fI,0 (θ) = lim fI,0 (θ),

θ→π −

θ→π −

θ→π −

since Ψφ − Ψ0 equals −φ (mod 2π), by (29) and (30). This proves the second statement; the proof of the first statement is entirely similar, if one takes into account the difference in orientation at N.

13

4

The Newman-Penrose Theory

In this section we review the spin s theory of Newman and Penrose [39]. We do so within the rigorous framework we have presented in section 3. Most of our arguments are very close to theirs. We resume presenting new results in the next section. Although we earlier used stereographic projection T from the north pole onto the tangent plane at S, in what follows we will follow Newman and Penrose and instead use stereographic projection σ from the north pole onto the equatorial plane, from UI ⊆ S 2 to C∗ := C \ {0}. Note that σ = T /2 is still orientation-preserving. Explicitly, if p ∈ UI has Euclidean coordinates (x, y, z), then σp =

1 (x + iy), 1−z

(31)

which one easily visualizes by using similar triangles. From this, and the formulas sin θ = 2 sin 2θ cos θ2 , 1 − cos θ = 2 sin2 θ2 , it follows that if p ∈ UI has standard spherical coordinates (θ, φ), then θ σp = eiφ cot( ). 2

(32)

The map σ, being a conformal map, preserves angles; we will also need to know the factor by which it multiplies lengths. Thus, if p ∈ UI , since σ is conformal, there is a number λp > 0 such that for any 1/2 tangent vector v at p, |σ ∗p v| = λp < v, v >E . We need to find the conformal factor λp . For this, we may assume that v = ρI (p) is a unit tangent vector tangent to the circle φ = constant through p. Restricted to that circle, by (31), the map σ is simply a dilation by the factor (1 − z)−1 , so we must have λp = (1 − z)−1. It is easy to calculate, then, from (31), that λp = 21 (1 + |σp|2 ). For this reason, for ζ ∈ C, we set 1 (33) P (ζ) = (1 + |ζ|2 ), 2 and we have shown that: 1/2

if p ∈ UI , σp = ζ, and v is a tangent vector at p, then |σ ∗p v| = P (ζ) < v, v >E .

(34)

We also remark the following fact (which is also geometrically evident). Since σ restricted to the aforementioned circle is just a dilation, if If p = (θ, φ) ∈ UI , then the angle from σ ∗ ρI (p) to ∂/∂y is − φ.

(35)

Indeed, it equals the angle from ρI (p) to ∂/∂y in R3 , which is surely −φ. We are now ready for the observations of Newman and Penrose, expressed in our language. These observations will be preceded by bullet points in what follows. We will provide rigorous proofs of their observations, within the context of spin s line bundles. Our proofs are modified versions of their arguments. Say Ω ⊆ UI is open. Suppose h(I) : Ω → C is smooth. Following Newman and Penrose we define ∂ i ∂ s (sin θ)−s h(I) . ∂sI h(I) = −(sin θ) + ∂θ sin θ ∂φ (Here we are again using standard spherical coordinates (θ, φ) on UI .) Thus ∂sI h(I) is a smooth function on Ω. Next, if h is a smooth section of LsΩ , we define ∂h to be the section of Ls+1 over Ω such that (∂h)I = ∂sI hI .

14 Also, if Ω0 ⊆ C∗ is open, and u is a smooth section of LsΩ0 , we define the smooth section Ds u of Ls+1 Ω0 by Ds u = 2P 1−s

∂P s u . ∂ζ

(Here P is as in (33). Recall that we have only one chart on C∗ , so sections of LsΩ0 or Ls+1 Ω0 may be naturally identified with smooth functions from Ω0 to C.) −1 If f is a section of LsΩ , where Ω ⊆ UI , let us write f σ ) = f ◦s σ −1 . The first key observation of Newman and Penrose is: • Suppose f is a smooth section of LsΩ , where Ω ⊆ UI is open. Then (∂f )σ

−1

= Ds (f σ

−1

).

(36)

To see this, one notes that ∂sI fI = s cot θfI − LfI , where L = (∂/∂θ) + (i/ sin θ)(∂/∂φ). Using (35), we see that the left side of (36) is e−i(s+1)φ [(s cot θ)fI ◦ σ −1 − (LfI ) ◦ σ −1 ]. Let u = fI ◦ σ −1 . A brief calculation, using (32), the chain rule, and the fact that cot(θ/2)/ sin θ = 12 csc2 (θ/2), shows that (LfI ) ◦ σ −1 = −eiφ csc2 (θ/2)∂u/∂ζ. On the other hand, if p = (θ, φ) ∈ UI and ζ = σp, then P (ζ) = 1 1 2 2 2 [1 + cot (θ/2)] = 2 csc (θ/2). Thus the left side of (36) equals −i(s+1)φ iφ ∂u . (37) e (s cot θ)u + 2P e ∂ζ On the other hand, the right side of (36) is 2P 1−s

∂e−isφ P s u , ∂ζ

(38)

which equals 2P 1−s [∂e−isφ P s /∂ζ]u + 2P e−isφ ∂u/∂ζ. We are left with showing that 2P 1−s

∂e−isφ P s = e−i(s+1)φ s cot θ. ∂ζ

(39)

Surely ∂P/∂ζ = 21 ζ. On the other hand, since e2iφ = ζ/ζ, we find 2ie2iφ ∂φ/∂ζ = 1/ζ, so that ∂φ/∂ζ = 1/(2iζ). Thus the left side of (39) equals e−isφ [−sP/ζ + sζ] = se−isφ [(|ζ|2 − P )/ζ] = se−i(s+1)φ [(|ζ|2 − P )/ cot(θ/2)], so all that is left to show is that (|ζ|2 − P )/ cot(θ/2) = cot θ. But (|ζ|2 − P )/ cot(θ/2) = [cot2 (θ/2) − 1]/[2 cot(θ/2)] = cot θ as desired. Next, Newman and Penrose observe: • Say R ∈ SO(3). Suppose f is a smooth section of LsΩ , where Ω ⊆ UI ∩ UR is open. Say R ∈ SO(3). Then ∂(f R ) = (∂f )R . (40) (Note that these sections are defined over R−1 Ω ⊆ UI .) −1 −1 To see (40), it suffices to show that [∂(f R )]σ = [(∂f )R ]σ . (Note that these sections are defined over −1 Ω1 := σ(R−1 Ω).) Set u = f σ , defined over Ω0 := σΩ, and let I = σ ◦ R ◦ σ −1 : Ω1 → Ω0 . Then by (36), [∂(f R )]σ

−1

= Ds (uσ◦R◦σ

−1

) = Ds uI .

(41)

15 On the other hand, [(∂f )R ]σ

−1

= (∂f )σ

−1

◦σ◦R◦σ −1

= (Ds u)I .

Thus, we need only show that if u is a section of LsΩ0 ⊆ C∗ , then Ds uI = (Ds u)I .

(42)

(Note that these sections are defined over Ω1 .) Now I is a conformal mapping. At any point ζ of Ω1 , if I ′ (ζ) = reiψ (ψ ∈ [−π, π)), then we know that the angle from I∗ζ (∂/∂y) to ∂/∂y is − ψ,

(43)

while for any real nonzero tangent vector v at ζ, |I∗ζ v| = r|v|. By (34) and (41), we conclude that I ′ (ζ) =

P (I(ζ)) iψ e . P (ζ)

(44)

Accordingly P (ζ)eiψ = Turning now to (42), we see by (45) that

P (I(ζ)) I ′ (ζ)

.

(45)

h i s ′ (ζ)s iψ s ∂ ([P u] ◦ I)(ζ)/I ∂ (P (ζ)e ) (u ◦ I)(ζ) 2P (ζ) = . (Ds uI )(ζ) = 2P (ζ)1−s ∂ζ P (ζ)s ∂ζ The key point is that I ′ (ζ)s is antiholomorphic, so that ∂([P s u] ◦ I)(ζ) 2P (ζ) (Ds u )(ζ) = = s ′ ∂ζ P (I(ζ))s e−isψ [P (ζ)I (ζ)] 2P (ζ)

I

∂(P s u) ◦ I(ζ) I ′ (ζ). ∂ζ

Finally, then, by (44), (Ds uI )(ζ) = 2P (I(ζ))1−s ei(s+1)ψ

∂(P s u) ◦ I(ζ) = (Ds u)I (ζ). ∂ζ

This proves (42), as desired. Next, say Ω ⊆ UR is open, so that R−1 Ω ⊆ UI . Say h(R) : Ω → C is smooth; then h(R) ◦ R : R−1 Ω → C is smooth, so we can apply ∂sI to it and obtain another smooth function on R−1 Ω. Now we define ∂sR h(R) = [∂sI (h(R) ◦ R)] ◦ R−1 ,

(46)

which is, like h(R) itself, a smooth function on Ω. We now deduce: • Suppose f is a smooth section of Ls over an open set Ω ⊆ UI ∩ UR . Then (∂f )R = ∂sR fR . To see this, say p ∈ Ω. Then by (28) and (40), (∂f )R (p) = [(∂f )R ]I (R−1 p) = (∂f R )I (R−1 p). Thus, by the definition of ∂sI and (28) again, (∂f )R (p) = (∂sI fIR )(R−1 p) = (∂sI [fR ◦ R])(R−1 p) = (∂sR fR )(p),

(47)

16 as desired. We can reformulate the last result more appealingly as follows. On the chart UR we again use coordinates (θ R , φR ), where if the (θ, φ) coordinates of p ∈ UI are (θ 0 , φ0 ), then the (θ R , φR ) coordinates of Rp ∈ UR are also (θ 0 , φ0 ). Then, in these coordinates on UR , ∂ i ∂ s ∂sR fR = −(sin θR ) (sin θR )−s fR . (48) + ∂θR sin θR ∂φR So far we have only defined ∂ over the chart UI , but (47) enables us to define ∂f for any smooth section f of LsΩ , where now Ω is any open subset of S 2 . We simply let ∂f be the smooth section of Ls+1 Ω , with the property that if p ∈ Ω is in UR , then (∂f )R (p) = ∂sR fR (p).

(49)

Note that this definition makes sense. Indeed, over Ω ∩ UI , this section agrees with the section ∂f we have previously defined. If, instead, p is the north or south pole, choose a particular R ∈ SO(3) with p ∈ UR , and let h be the section of Ls over Ω ∩ UR such that hR = ∂sR fR . Then on Ω ∩ UR ∩ UI , h agrees with ∂f as previously defined. Thus, for any other rotation R′ such that p ∈ UR′ , hR′ (q) = ∂sR′ fR′ (q) for all q ∈ Ω ∩ UR ∩ UR′ ∩ UI . Letting q → p, we see that hR′ (p) = ∂sR′ fR′ (p) as well. This shows that the definition (49) makes sense. As a consequence we have that: ∂ maps smooth sections of Ls to smooth sections of Ls+1 .

(50)

Here the sections could be defined over the entire sphere; ∂ is consequently called the “spin-raising operator”. Note also that, in (40), we may now relax the hypothesis that Ω ⊆ UI ∩ UR , and only assume that Ω ⊆ S 2 is open. Indeed, both sides of (40) agree on Ω ∩ UI ∩ UR ; since both sides are smooth sections of LsΩ , they must agree on all of Ω. As is customary for sections of line bundles, we let C ∞ (Ls ) denote the space of smooth sections of Ls , defined over all of S 2 . (In section 2, we called this space Cs∞ (S 2 ).) Newman and Penrose make the following important observation: • If s < 0, then ∂ : C ∞ (Ls ) → C ∞ (Ls+1 ) is injective.

(51)

To see this, say f ∈ C ∞ (Ls ), and that ∂f = 0. By Theorem 3.1, |fI | is bounded on UI . Let u = fI ◦ σ−1 ; then u is bounded on C∗ . As in (36) and (38), we see that 0 = (∂f )σ

−1

= Ds (f σ

−1

) = 2P 1−s [∂e−isφ P s u/∂ζ].

The function v := eisφ P s u is therefore holomorphic on C∗ . It is bounded near 0, so it extends holomorphically to C. But, since s < 0, [P (ζ)]s → 0 as ζ → ∞, v ≡ 0 by Liouville’s theorem. Consequently u, and hence f , are both zero, as desired. Say now that Ω ⊆ S 2 is open. There is evidently a well-defined map of complex conjugation from C ∞ (LsΩ ) to C ∞ (L−s Ω ), taking a section f in the former space to a section f in the latter space, defined by (f )R (p) = fR (p) for all p ∈ UR ∩ Ω (for any R ∈ SO(3)). We may then define the “spin-lowering operator” ∂ by ∂f = ∂f , s−1 so that ∂ : C ∞ (LsΩ ) → C ∞ (LΩ ). In the coordinates (θ R , φR ) on UR , we evidently have that

(∂f )R = ∂ sR fR ,

(52)

17 where ∂ sR fR = −(sin θR )−s

∂ i ∂ − ∂θR sin θR ∂φR

(sin θR )s fR .

(53)

s−1 by If Ω0 ⊆ C∗ is open, and u is a smooth section of LsΩ0 , we define the smooth section Ds u of LΩ 0 Ds u = Ds u; thus ∂P −s u Ds u = 2P 1+s . ∂ζ

By taking complex conjugates in (36), (40), and (51), we find: • Suppose f is a smooth section of LsΩ , where Ω ⊆ UI is open. Then (∂f )σ

−1

= Ds (f σ

−1

).

(54)

• Suppose f is a smooth section of LsΩ , where Ω ⊆ S 2 is open. Say R ∈ SO(3). Then ∂(f R ) = ∂f R . • If s > 0, then ∂ : C ∞ (Ls ) → C ∞ (Ls−1 ) is injective. 2

(55) (56)

s

One can define L sections of L , (or on any other line bundle over a compact oriented manifold, for that matter), and define an L2 norm on them, by using a finite atlas on the bundle and a partition of unity on the manifold. Different choices of atlas and partition of unity lead to the same space of L2 sections with an equivalent L2 norm. For the bundle Ls there is a very natural choice of L2 norm, equivalent to the L2 norm obtained through such a construction. Note first that if f is an L2 section of Ls , then (a representative of) f is defined almost everywhere. Say f is defined at p ∈ UR ∩ UR′ . We may then write ΦR (f (p)) = (p, fR (p)), ΦR′ (f (p)) = (p, fR′ (p)), fR′ (p) = eisψ fR (p), where ψ is the angle from ρR (p) to ρR′ (p). Thus |fR (p)|2 is independent of the choice of R, as long as p ∈ UR . Accordingly |f (p)|2 := |fR (p)|2 (p ∈ UR ) is well-defined a.e. on S 2 . We then define the L2 norm of f by Z 2 |f |2 kf kL2 = S2

where the integral is with respect to the usual surface measure on S 2 . We let L2 (Ls ) denote the space of L2 sections of Ls , with this norm. We may now show: • The formal adjoint of ∂ : C ∞ (Ls ) → C ∞ (Ls+1 ) is −∂. For this, we need to show that, whenever f ∈ C ∞ (Ls ) and g ∈ C ∞ (Ls+1 ), we have < ∂f, g >= − < f, ∂g >,

(57)

where < ·, · > denotes L2 inner product. For this, we first note the following useful proposition: Proposition 4.1 (a) Let N be a positive integer. Then S 2 is not the union of N open subsets of diameter less than π/N . (b) Any open set Ω ⊆ S 2 of diameter less that π/2 is contained in some UR .

18 Proof For (a), say it were, and let U1 be one of those open subsets. By connectedness of S 2 , U1 must intersect another one of those sets, say U2 . Again by connectedness, U1 ∪ U2 must intersect another one of those sets, say U3 . Continuing in this manner, we obtain the contradiction π = diamS 2 ≤ diamU1 + diamU2 + . . . < N π/N.

For (b), we need to show that Ω omits some pair of antipodal points in S 2 . If it did not, then if A is the antipodal map, Ω ∪ AΩ = S 2 . This contradicts (a). Let us then cover S 2 by a finite collection of open sets P of diameter lessPthan π/5, choose a partition of unity {ζ j } subordinate to this cover, and break up f = ζ j f and g = ζ j g. By Proposition 4.1 (b), by using this breakup, in proving (57), we may assume that suppf ∪ suppg ⊆ UR for some R. Then we need to show that Z 2π Z π Z 2π Z π [fR ∂ s+1,R gR ](θR , φR ) sin θ R dθ R dφR . [(∂sR fR )gR ](θ R , φR ) sin θR dθR dφR = − 0

0

−π

−π

But this is an elementary calculation, using (48) and (53). We are now ready (again following Newman and Penrose) to define the spin s spherical harmonics. We let {Ylm : 0 ≤ l, −l ≤ m ≤ l} be the usual orthonormal basis of spherical harmonics for S 2 , Ylm (θ, φ) = eimφ (−1)m

−

X alm sin2l (θ/2) l! r

l r

l r−m

where

(−1)l−r cot2r−m (θ/2).

(58)

1

alm = [(l + m)!(l − m)!(2l + 1)/(4π)] 2 , and our convention is that the binomial coefficient nj is zero if j < 0 (or n < 0). (As usual m+ = − max(m, 0), m− = max(−m, 0). The factor of (−1)m in (58) is usually not included, but by including it, we have Ylm = Yl,−m , which will simplify many formulas in the sequel.) For l ≥ |s|, Newman and Penrose then define the spin s spherical harmonic s Ylm by h i1   (l−s)! 2 ∂ s Ylm (0 ≤ s ≤ l), (l+s)! (59) i 12 h s Ylm =   (l+s)! (−∂)−s Ylm (−l ≤ s ≤ 0). (l−s)!

The s Ylm are not defined for l < |s|. Note that s Ylm is a smooth section of Ls . We have ([39], [22]): s YlmI (θ, φ)

= eimφ (−1)m

−

X alm sin2l (θ/2) 1/2 [(l + s)!(l − s)!] r

l−s r

l+s r+s−m

(−1)l−r−s cot2r+s−m (θ/2).

(60) To see this, let s vlm denote the section of Ls over UI such that s vlmI equals the right side of (60). We σ −1 need to show that s vlm = s Ylm over UI . Note that this is surely true when s = 0. Let s ulm = s vlm . By (36), when s ≥ 0, we need only show that Ds 0 ulm =

(l + s)! (l − s)!

21

s ulm .

(61)

(Here Ds = Ds−1 ◦ . . . ◦ D0 .) However, using (35) again, one easily calculates that, whenever l ≥ |s|, +

X (−1)l−m alm 2 −l (1 + |ζ| ) s ulm (ζ) = [(l + s)!(l − s)!]1/2 r

l−s r

l+s r+s−m

ζ r (−ζ)r+s−m .

(62)

19 From this, a brief calculation using Pascal’s rule and the identities l−s = (r + 1) r+1 = (l − s) (l − s − r) l−s r

l−s−1 r

shows that, whenever l ≥ |s|,

1

Ds s ulm (ζ) = [(l − s)(l + s + 1)] 2

s+1 ulm .

(63)

(Here, our convention is, that since the factor (l − s) appears on the right side of (63), the right side is to be interpreted as zero if l = s.) From this, (61) follows at once by induction on s, for s ≥ 0, and so we have (60) for s ≥ 0 as well. We then obtain (60) for s < 0 by complex conjugation and use of the identity s YlmI

=

−s Yl,−m,I .

Even better, by (63) and (36), we have that, whenever l ≥ |s|, 1

∂ s Ylm (ζ) = [(l − s)(l + s + 1)] 2

s+1 Ylm .

(64)

By taking complex conjugates of this, we also find that, whenever l ≥ |s|, 1

∂ s Ylm (ζ) = − [(l + s)(l − s + 1)] 2

s−1 Ylm .

Newman and Penrose next observe that, for each s, the {s Ylm } are orthonormal. Indeed, set ( [(l + s)!/(l − s)!]1/2 if 0 ≤ s ≤ l, bls = [(l − s)!/(l + s)!]1/2 if − l ≤ s ≤ 0. By (64), if s > 0, s Ylm

=

while if s < 0, s Ylm

If s ≥ 0, we see that < s Ylm , s Yl′ m′ >=

=

1 s ∂ Ylm , bls

Ylm =

1 (−∂)−s Ylm , bls

1 (−∂)s s Ylm , bls

Ylm =

1 −s ∂ s Ylm . bls

(65)

(66)

(67)

(68)

1 1 < ∂ s Ylm , s Yl′ m′ >= < Ylm , (−∂)s s Yl′ m′ >=< Ylm , Yl′ m′ >= δ ll′ δ mm′ , bls bls

which proves the orthonormality. If s < 0, one obtains the orthonormality through complex conjugation. Thus, the { s Ylm } are an orthonormal set of eigenfuntions for the (formally) positive self-adjoint operator −∂∂ : C ∞ (Ls ) → C ∞ (Ls ). Since this operator is elliptic (by (48) and (53)), it has an orthonormal basis of smooth eigenfunctions, so at this point one would surely expect: • The { s Ylm } are an orthonormal basis of smooth eigenfunctions for − ∂∂ : C ∞ (Ls ) → C ∞ (Ls ). (69) This would follow if we knew: • If s > 0, then ∂ s : C ∞ → C ∞ (Ls ) is surjective.

(70)

Indeed, say s > 0. Now ∂ s : C ∞ → C ∞ (Ls ) is continuous (using their Fréchet space topologies). (As usual, this topology is obtained by using a finite atlas on the bundle and a partition of unity on the manifold. Different choices of atlas and partition of unity lead to an equivalent Fréchet space topology.) Thus, since the linear span of the {Ylm } is dense in C ∞ , the linear span of the { s Ylm } is dense in ∂ s C ∞ = C ∞ (Ls ), which is in turn L2 -dense in the space of L2 sections. This would prove (69) for s > 0 (and hence for all s by complex conjugation), if we knew (70). Moreover, (70) itself follows from the s injectivity of ∂ : C ∞ (Ls ) → C ∞ (see (56)), and by use of elliptic theory and Sobolev spaces. Rather than proceed in this manner, and in order to extract important additional information, we give a direct proof of (69) and (70), by following the method of Newman and Penrose. Their method was somewhat formal, but it can easily be made rigorous. In fact, we shall show:

20 Theorem 4.2 Fix an integer s. (a) Let {k kN } be a family of norms defining the Fréchet topology on C ∞ (Ls ). Then for any N there exist C, M such that ks Ylm kN ≤ C(l + 1)M (71) whenever l ≥ |s|. (b) For every f ∈ C ∞ (Ls ), there exist numbers {Alm : l ≥ |s|, −l ≤ m ≤ l} such that XX f= Alm s Ylm

(72)

l≥|s| m

(convergence in C ∞ (Ls )), such that the Alm decay rapidly, by which we mean that for every N there exist C, M such that |Alm | ≤ C(l + 1)−M (73) for all l, m. Further, we must have Alm =< f, s Ylm >. Proof (a) is well-known when s = 0. It then follows for all s by use of (59), as well as (53) and (48) for general R. Note, then, that any series as on the right side of (72), where the Alm decay rapidly, does converge in C ∞ (Ls )). (b) is also well-known when s = 0. To prove it in general, we may as usual assume s > 0. In that case we may write XX XX s ∂ f= Blm Ylm = Blm Ylm + g l≥0 m

l≥s m

P

P

where the Blm decay rapidly, and where g = l, by the orthonormality of the s Ylm . This completes the proof. Moreover, (70) follows as well, since if s > 0, XX f = ∂ s( Clm Ylm ), l≥s m

if Clm = Alm /bls . As we have said, the completeness of the {s Ylm } as an orthonormal basis follows from (70). It also follows directly from Theorem 4.2 (b), which shows that the linear span of the { s Ylm } is dense in C ∞ (Ls ). If s > 0, we may write M C∞ = V W (75) where

s

V = ∂ C ∞ (Ls ) = hYlm : l ≥ si

(76)

W = ker∂ s = hYlm : l < si.

(77)

∂ s : V → C ∞ (Ls ) bijectively.

(78)

and Moreover,

21 The statements (75) and (78) are evident, once one identifies V and W as spaces spanned by certain spherical harmonics, as in (76) and (77). One could obtain (75) and (78) by using only the first definitions of V and W in (76) and (77), by using elliptic theory and (56). However, by identifying them as spaces spanned by certain spherical harmonics, Newman and Penrose observe a crucial additional structure. Namely, the decomposition (75) respects the splitting of a function into its real and imaginary parts. That is, if g ∈ V (or W ), so are ℜg and ℑg. (This is evident, since Ylm = Yl,−m .) This splitting of any function in V then induces, through the isomorphism in (78), a corresponding splitting of any element of C ∞ (Ls ). Thus, say f ∈ C ∞ (Ls ). Suppose s > 0. We write f = ∂ s g for some g ∈ C ∞ , then set fE = ∂ s ℜg, fM = ∂ s ℑg. fE and fM are well-defined, independent of choice of g. Indeed, if ∂ s h = 0, then h ∈ W , so ℜh and ℑh are in W , so ∂ s ℜh = ∂ s ℑh = 0. In particular, we could let g be the unique solution of f = ∂ s g in V . s Similarly, if s < 0, we write f = ∂ g for some g ∈ C ∞ , then set s

s

fE = ∂ ℜg, fM = ∂ ℑg. Surely we always have f = fE + ifM . Again suppose s > 0. By repeated use of (55), one sees that the space V is rotationally invariant, and then by repeated use of (40), one sees that (f R )E = (fE )R , (f R )M = (fM )R ,

(79)

for any R ∈ SO(3). Similarly these relations hold for s < 0 as well. In fact, if f is as in (72), it is very easy to find the expansions of fE and fM in terms of the s Ylm . Say first s > 0. The solution g of f = ∂ s g, with g ∈ V , is simply X X Alm l≥s m

bls

Ylm .

A very brief calculation, using Ylm = Yl,−m , now shows that XX fE = AlmE s Ylm

(80)

l≥|s| m

and fM =

XX

AlmM s Ylm

(81)

AlmE = (Alm + Al,−m )/2,

(82)

AlmM = −i(Alm − Al,−m )/2.

(83)

l≥|s| m

where while Similarly, (80), (81), (82) and (83) hold for s < 0 as well. Up to normalization constants, these formulas agree with formulas in section II of [53], and also with equation (54) of [7] (to see the latter, use equations (55), (65), (66) and (69) of that article). (Those articles take s = 2 and write f = Q + iU . [7] uses the C notation aG (lm) and a(lm) in place of our AlmE and AlmM .) Let us now let ∆s =

(

−∂∂ −∂∂

if s ≥ 0, if s < 0.

(84)

22 (Here ∆s acts on smooth sections of Ls .) We also let ( (l − s)(l + s + 1) λls = (l + s)(l − s + 1)

if 0 ≤ s ≤ l, if − l ≤ s < 0.

(85)

Hls =< s Ylm : −l ≤ m ≤ l > (for l ≥ |s|).

(86)

Pls denotes the projection in L2 (Ls ) onto Hls .

(87)

We also let Also L

Then L2 (Ls ) = l≥|s| Hls . We also have that Hls is the space of smooth sections of Ls which are eigenfunctions of ∆s for the eigenvalue λls . (This follows from (64), (65), and Theorem 4.2.) (In particular, then, ∆0 is the usual spherical Laplacian on S 2 .) This was all noted by Newman and Penrose. −1

Of course {s Ylm : −l ≤ m ≤ l} is an orthonormal basis for Hls . Note that the map f → f R = Rs ◦f ◦R−1 R−1 is a unitary mapping of L2 (Ls ) onto itself, for any R ∈ SO(3). Note also that for each such R, s Ylm is an eigenfunction of ∆s for the eigenvalue λls . (This follows from (40) and (55).) Thus, for any R, R−1 {s Ylm : −l ≤ m ≤ l} is also an orthonormal basis of Hls . Note also that, by (28), we have, for p ∈ UR , −1

R (s Ylm )R (p) = s YlmI (R−1 p).

(88)

and so, in the (θ R , φR ) coordinates on UR , we have −1

R (s Ylm )R (θ R , φR ) = eimφR (−1)m

5

−

X alm sin2l (θR /2) 1/2 [(l + s)!(l − s)!] r

l−s r

l+s r+s−m

(−1)l−r−s cot2r+s−m (θ R /2). (89)

Zonal Harmonics and Kernels

We now return to our new results. Concerning the s Ylm , we also make the following observation, which is essentially in [7], section 6. Proposition 5.1 lim e

θ→0+

isφ

s YlmI (θ, φ)

=

(

+

(−1)s 0

2l+1 1/2 4π

if m = −s otherwise

Proof Since s YlmI (θ, φ) equals eimφ times a function of θ, this limit (which surely exists, by Theorem 3.1) must be zero if m 6= −s. If m = −s, it follows from an examination of (58). (In each term in the summation in (60), because of the binomial coefficients, we must have r ≤ l − s. As θ → 0+ , sin2l (θ/2) cot2r+s−m (θ/2) = sin2l (θ/2) cot2r+2s (θ/2) has a nonzero limit only if 2l ≤ 2r + 2s. These two conditions together allow us to look only at the term in which r = l − s when taking the limit, and the proposition now follows at once.) We next discuss the kernel of the projection operator Pls ; we claim that it is a smooth-kernel operator. Since Pls maps sections in L2 (Ls ) to smooth sections of Ls , we must define what we mean by the kernel of such an operator. For x ∈ S 2 , let Lsx be π −1 (x), the fiber above x. Definition 5.2 We say that T : L2 (Ls ) → C ∞ (Ls ) is a smooth-kernel operator if T has the form Z K(x, y)f (y)dS(y) (90) (T f )(x) = S2

23 for any f ∈ L2 (Ls ); here we require: (i) For each (x, y) ∈ S 2 × S 2 , K(x, y) : Lsy → Lsx is linear; (ii) K(x, y) depends smoothly on (x, y) ∈ S 2 × S 2 . We also call the restriction of T to C ∞ (Ls ) a smooth-kernel operator. We call the collection of linear maps K = {K(x, y) : (x, y) ∈ S 2 × S 2 } the kernel of T . Note here that the integral in (90) is a vector-valued integral. Also note that the statement (ii) means that the kernel is smooth in (x, y) after one locally trivializes the bundle. To say that T is a smooth-kernel operator is evidently equivalent to saying the following: for each R, R′ ∈ SO(3), there exists a function KR′ ,R ∈ C ∞ (UR′ × UR ), such that for all smooth sections f of Ls with compact support in UR , we have Z KR′ ,R (x, y)fR (y)dS(y), (91) (T f )R′ (x) = S2

for all x ∈ UR′ . Evidently, if R, R′ , R1 , R1′ ∈ SO(3), if (91) holds, we must have KR1 ′ ,R1 (x, y) = eis(ψ 1 −ψ2 ) KR′ ,R (x, y)

(92)

for all x ∈ UR1 ′ ∩ UR′ and y ∈ UR1 ∩ UR , where ψ 1 is the angle from ρR′ (x) to ρR1 ′ (x), and ψ 2 is the angle from ρR (y) to ρR1 (y).

(93)

Let us now examine the kernel of the projection operator Pls . Evidently, if f ∈ L2 (Ls ), we have Pls f =

l X

< f, s Ylm > s Ylm .

(94)

m=−l R−1

(Here one could use s Ylm0 in place of s Ylm , for any R0 ∈ SO(3).) Thus Pls has kernel K ls , where for any R, R′ ∈ SO(3), ls KR ′ ,R (x, y) =

l X

s YlmR′ (x)s YlmR (y).

(95)

m=−l R−1

(Again here, one could use s Ylm0 in place of s Ylm , for any R0 ∈ SO(3).) Definition 5.3 For |l| ≥ s, we define s Zl , the “s-zonal harmonic” in Hls , by +

s s Zl = (−1)

2l + 1 4π

1/2

s Yl,−s .

(96)

We have the following new theorem, which provides justification for this definition. When s = 0 this theorem reduces to the well-known result that 0 Zl (x)

:= Zl (x) = K l0 (x, N) is the zonal harmonic in Hl0 .

(97)

24 Theorem 5.4 (a) Say N ∈ UR , and that x ∈ UI ∩ UR′ . Then s ZlI (x)

ls = eis(ψ1 −ψ2 ) KR ′ ,R (x, N),

(98)

where ψ 1 is the angle from ρR′ (x) to ρI (x), and ψ 2 is the angle from ρR (N) to ∂/∂y (at N). (b) For any f ∈ Hls , we have

lim eisφ fI (θ, φ) =< f, s Zl > .

(99) (100)

θ→0

ls Proof For (a), note that if y ∈ UI ∩ UR , then KI,I (x, y) equals

X

s YlmI (x)s YlmI (y)

ls = eis(ψ 1 −ψ2 ) KR ′ ,R (x, y)

m

where ψ 1 is the angle from ρR′ (x) to ρI (x), and ψ 2 is the angle from ρR (y) to ρI (y). In this equation, let us say y has coordinates (θ, φ) (in the usual coordinates on UI ). Let us fix φ, multiply both sides of the equation by e−isφ , and take the limit as θ → 0+ . (One can, if one wishes, choose to take φ = 0; or one can let φ be arbitrary, and recall our choice of orientation at N.) (a) now follows at once from Proposition 5.1. To prove (b), one may assume f = s Ylm for some m; but then the result is evident from Proposition R ls (x, y)fI (y)dS(y) (which is valid for 5.1. (Alternatively, one can start with the equation fI (x) = S 2 KI,I f ∈ Hls , x ∈ UI ), multiply by eisφ , and take a limit.) This completes the proof. ls For another new result, also familiar when s = 0, we examine KR,R (x, x), for x ∈ UR . It is evident from (92), (93) that this quantity is independent of R. In fact it is independent of x as well:

Theorem 5.5 If x ∈ UR , then

ls (x, x) = KR,R

2l + 1 . 4π

(101)

Proof It is enough to prove this for R = I, since we will then know it unless x = N or S, and the result will follow in those cases too by continuity. Let us then take R = I and free the letter R for other uses. Say now x, y ∈ UI . Choose R ∈ SO(3) with Ry = x. Then x ∈ UI ∩ UR , so X X ls ls R−1 ls KI,I (x, x) = KR,R (x, x) = |(s Ylm )R (x)|2 = |s YlmI (R−1 x)|2 = KI,I (y, y). m

m

ls (We used (88).) Accordingly KI,I (x, x) is independent of x ∈ SO(3). Thus for any fixed x ∈ SO(3), Z XZ ls ls KI,I (y, y)dS(y) = |s Ylm (y)|2 dS(y) = 2l + 1, 4πKI,I (x, x) = S2

m

S2

as desired. In doing estimates involving the K ls , it will be very useful to observe the theorem which follows. In [s] [s] this theorem, ∂R will denote the differential operator on UR which satisfies (∂ s F )R = ∂R FR , for any [s]

[s]

ordinary smooth function F on UR ; explicitly, ∂R = ∂s−1,R ◦ . . . ∂0,R . Similarly, ∂ R will denote the s

[s]

differential operator on UR which satisfies (∂ F )R = ∂ R FR , for any ordinary smooth function F on UR . Theorem 5.6 Say x ∈ UR′ and y ∈ UR . Then if s > 0, ls KR ′ ,R (x, y) =

(l − s)! [s] [s] l0 ∂ ′ ∂ K (x, y), (l + s)! R ,x R,y

(102)

25 while if s < 0, ls KR ′ ,R (x, y) = [s]

[s]

(l + s)! [−s] [−s] l0 ∂ ′ ∂ K (x, y). (l − s)! R ,x R,y

(103)

(Here ∂R′ ,x denotes ∂R′ taken in the x variable, etc.) Proof This follows at once from (67), (68), and (95). 1/2

Of course K l0 (x, y) = Zl,x (y), the zonal harmonic in Hl0 based at x. It equals [(2l + 1)/4π]Pl (x · y), where Plλ is the ultraspherical (or Gegenbauer) polynomial of degree l associated with 1/2. Series defining wavelets on the sphere, involving the K l0 , were estimated in [37], [38] and [18]. Using these estimates, and Theorem 5.6, we will be able to estimate similar series, which involve the K ls for general s, in place of the K l0 , in the next section.

6

Spin Wavelets

In sections 3-5 we have often used the variable “f ” to denote a spin function on the sphere. In the remainder of this article, the variable “f ” will be reserved for another purpose. Specifically, say f ∈ S(R+ ), f 6= 0, and f (0) = 0. Let (M, g) be a smooth compact oriented Riemannian manifold, and let ∆ be the Laplace-Beltrami operator on M (for instance, the spherical Laplacian if M is S 2 ). Let Kt be the kernel of f (t2 ∆). Then, as we have explained in section 2, the functions wt,x (y) = K t (x, y),

(104)

if multiplied by appropriate weights, can be used as wavelets on M. In case M = S n and f has compact support away from the origin, we shall say these wavelets are “needlet-type”. On the sphere S 2 , we have l ∞ X X f (t2 λl )Ylm (x)Ylm (y). Kt (x, y) = (105) l=0 m=−l

As we have explained in section 2, a key property of this kind of wavelet is its localization. Working in the general situation (general M, general f ), one has: for every pair of C ∞ differential operators X (in x) and Y (in y) on M, and for every nonnegative integer N , there exists c := cN,X,Y such that for all t > 0 and x, y ∈ M |XY Kt (x, y)| ≤ c

t−(n+I+J) , (d(x, y)/t)N

(106)

where I := deg X and J := deg Y . In the case where M is the sphere, f has compact support away from 0, and if one replaces λl = l(l + 1) by l2 in the formula for Kt , this was shown by Narcowich, Petrushev and Ward, in [37] and [38]. The general result was shown in [18]. It will be important to note that in (106), the constant c can be chosen to be ≤ Ckf kM , for some M . (Here we have chosen a nondecreasing family of norms {k kM } to define the Fréchet space topology on S0 (R+ ) = {f ∈ S(R+ ) : f (0) = 0}.) Indeed, this fact may be read off from the proof of (106). More simply, one may use the closed graph theorem: The collection of functions Kt (x, y) (smooth in t, x, y for t > 0) which satisfy (106) is itself naturally a Fréchet space, say F . We want to see that the map taking f to the function Kt,f (x, y) ∈ F is continuous. (Here we are temporarily letting Kt,f be the kernel of f (t2 ∆).) For this, by the closed graph theorem, one need only note that the map from S0 (R+ ) to C which takes f to Kt,f (x, y) (for any fixed t, x, y) is continuous; but this is evident from (105). The proof of (106) in [18] would in fact go through without change if, instead of Kt being the kernel of f (t2 ∆), where ∆ is the Laplace-Beltrami operator, one assumed that Kt was the kernel of f (t2 D), where D is any smooth positive second-order elliptic differential operator, acting on C ∞ (M). It is natural to

26 conjecture then that something analogous holds if D is instead a smooth positive second-order elliptic differential operator, between sections of line bundles on M. In fact, the following theorem says that this is the case for D = ∆s : C ∞ (Ls ) → C ∞ (Ls ): Theorem 6.1 Let s be an integer. Suppose f ∈ S(R+ ), f (0) = 0. Let Kt be the kernel of f (t2 ∆s ). Then: For every R, R′ ∈ SO(3), every pair of compact sets FR ⊆ UR and FR′ ⊆ UR′ , and every pair of C ∞ differential operators X (in x) on UR′ and Y (in y) on UR , and for every nonnegative integer N , there exists c such that for all t > 0, all x ∈ FR′ and all y ∈ FR , we have |XY Kt,R′ ,R (x, y)| ≤ c

t−(n+I+J) , (d(x, y)/t)N

(107)

where I := deg X and J := deg Y . Note The “needlet-type” case, where f has compact support away from the origin, is easiest, and that is the only case we will deal with in this article. The general case will be dealt with elsewhere. Proof We have X f (t2 λls )Pls (108) f (t2 ∆s ) = l≥|s|

(strong convergence), so that Kt,R′ ,R (x, y) =

X

ls f (t2 λls )KR ′ ,R .

(109)

l≥|s|

(The interchange of order of integration and summation, needed to derive (109), is easily justified through the rapid decay of f , and through (71), once one recalls (95).) Now, say s > 0. Then by Theorem 5.6, [s]

[s]

Kt,R′ ,R (x, y) = ∂R′ x ∂ Ry

X

f (t2 λls )

l≥|s|

(l − s)! l0 K (x, y), (l + s)!

(110)

(The interchange of order of differentiation and summation, needed to derive (110), is easily justified through the rapid decay of f , and through (71) in the case s = 0, once one recalls (95) in the case s = 0.) Now notice that for n ∈ Z, we have (l + n)(l + 1 − n) = l(l + 1) − γ n

(111)

where γ n = n(n − 1). Also note that (l + s)! = [(l + s)(l + 1 − s)][[(l + s − 1)(l + 1 − (s − 1))] . . . [(l + 1)(l + 1 − 1)]. (l − s)! Accordingly

Note also that

(l + s)! = [l(l + 1) − γ s ][l(l + 1) − γ s−1 ] . . . [l(l + 1) − γ 1 ]. (l − s)!

(112)

λls = l(l + 1) − γ −s = l(l + 1) − s(s + 1).

(113)

To prove the theorem, let us first note that if we restrict to t ∈ [T0 , T1 ], where 0 < T0 < T1 are fixed, then the result is trivial (even if we do not assume f has compact support away from the origin). Indeed, since d(x, y) ≤ 2π for all x, y, it is enough to show that XY Kt,R′ ,R is bounded for (t, x, y) ∈ [T0 , T1 ]×FR′ ×FR . This, however, is evident from (71) and the rapid decay of f .

27 For other t, we shall assume that the support of f is contained in some interval [a, b] with b > a > 0. In that case, there is a T1 > 0 such that Kt,R,R′ ≡ 0 for all t > T1 (indeed, this will be true if λss T12 > b.) Thus we need only prove that there exists a T0 > 0 such that the result holds for all t ∈ (0, T0 ]. In fact, choose any T0 > 0 with γ s T02 < a/2. For 0 < t ≤ T0 , define ft (u) =

[u − γ s

f (u − s(s + 1)t2 ) . − γ s−1 t2 ] . . . [u − γ 1 t2 ]

t2 ][u

(114)

The function ft is supported in [a, b + s(s + 1)T02 ], a fixed compact interval. (Note that the denominator in (114) does not vanish for u in this interval; in fact, each factor is at least a/2.) Then, in (110), by (112) we may write X l≥s

f (t2 λls )

X (l − s)! l0 K (x, y) = t2s ft (t2 λl )K l0 (x, y) = t2s K[t] (x, y) (l + s)!

(115)

l≥s

where K[t] is the kernel of ft (t2 ∆). Now, it is easy to see that the functions ft (t ∈ (0, T0 ]) form a bounded subset of S0 (R+ ). Thus, by the remarks following (106), for every pair of C ∞ differential operators X ′ (in x) and Y ′ (in y) on S 2 , and for every nonnegative integer N , there exists c such that whenever 0 < t ≤ T0 , and x, y ∈ S 2 , −(n+I ′ +J ′ ) ′ ′ X Y K[t] (x, y) ≤ c t , (116) (d(x, y)/t)N

where I ′ := deg X ′ and J ′ := deg Y ′ . In fact, this remains true if we insist x ∈ FR′ , y ∈ FR , and allow X ′ (resp. Y ′ ) to be smooth differential operators only in a neighborhood of FR′ (resp. FR ), since there are smooth differential operators on all of S 2 which agree with these on FR′ (resp. FR ). In the situation [s] of (107), note that ∂R′ x is a smooth differential operator of degree s in x in a neighborhood of FR′ , and [s]

∂ Ry is a smooth differential operator of degree s in y in a neighborhood of FR′ . Thus we have

t−[n+(I+s)+(J+s)] t−[n+I+J] [s] [s] |XY Kt,R′ ,R (x, y)| = t2s XY ∂R′ x ∂ Ry K[t] (x, y) ≤ ct2s = c , (d(x, y)/t)N (d(x, y)/t)N

as desired. Similar arguments work if s < 0. This completes the proof.

Remark 6.2 As in the case s = 0, the closed graph theorem implies that the constant P c in (107) can be chosen to be ≤ Ckf kM , for some M . (Here, for f ∈ S(R+ ), let us define kf kM = i+j≤M kxi ∂ j f k, where k k denotes sup norm.) Also, C and M depend only on I, J, N, R′ and R.) Note, then, that (107) remains true if Kt is the kernel of f (t2 ∆s ), where now only f ∈ S0M (R+ ) = {f ∈ C M (R+ ) : f (0) = 0 and kf kM < ∞}. This is evident, since S0 (R+ ) = ∩L S0L (R+ ) is dense in S0M (R+ ). Again we will can choose c to be ≤ Ckf kM , where C and M depend only on I, J, N, R′ and R. Let us now explain what we shall mean by spin wavelets. Say f ∈ S(R+ ), f 6= 0, f (0) = 0. Say s is an integer, R ∈ SO(3), t > 0, and x ∈ UR . Let us define XX f (t2 λls ) s YlmR (x) s Ylm . wtxR = (117) l≥|s| m

Then wtxR ∈ C ∞ (Ls ). Also note that wtxR,R′ (y) = K t,R,R′ (x, y)

(118)

2

(if x ∈ UR , y ∈ UR′ , and Kt is the kernel of f (t ∆s )). This generalizes the case s = 0 of (104). Moreover, if F ∈ L2 (Ls ), then for x ∈ UR β t,F,x,R := hF, wtxR i = (f (t2 ∆s )F )R (x) = (β t,F )R (x)

(119)

28 where we have set β t,F = (f (t2 ∆s )F ). P P 2 s l≥|s| m alm s Ylm ∈ L (L ), then

We call β t,F,x,R a spin wavelet coefficient of F .

β t,F =

XX

f (t2 λls )alm s Ylm (x).

If F =

(120)

l≥|s| m

so that β t,F,x,R =

XX

f (t2 λls )alm s YlmR (x).

(121)

l≥|s| m

In cosmological applications, one assumes that f is real-valued. In that case, by (120 and (80) – (83), we have that (122) β t,FE = [β t (F )]E ; β t (FM ) = [β t (F )]M for all F ∈ L2 (Ls ). Let us now explain how, in analogy to the case s = 0, one can obtain a nearly tight frame from spin wavelets. Let P = P|s|,s be the projection onto the H|s|,s , the null space of ∆ls (in C ∞ (Ls ). Then for a > 1 sufficiently close to 1, for an appropriate discrete set {xj,k }(j,k)∈Z×[1,Nj ] on M, certain Rj,k with xj,k ∈ URj,k , and certain weights µj,k , the collection of {φj,k := µj,k waj ,xj,k ,Rj,k } constitutes a nearly tight wavelet frame for (I − P )L2 (Ls ). (Note that this property is independent ofP the choice of Rj,k .) In that case, we will have that for some C, if F ∈ (I − P )L2 (Ls ), then, in L2 , F ∼ C j,k hF, φj,k iφj,k (note that this sum is independent of the choice of Rj,k ). We need to make this all precise. In order to find suitable xj,k , we need to subdivide S 2 into a fine grid. To this end, we select c0 , δ 0 > 0 with the following properties: (*) Whenever 0 < τ < δ, we can write S 2 as a finite disjoint union of measurable sets Ek such that the diameter of each Ek is less than or equal to τ , and such that the measure of each Ek is at least c0 τ 2 . It is easy to see that such τ , δ exist. (We could even require that each set Ek be either a spherical cap, or a spherical rectangle bounded by latitude and longitude lines. The existence of c0 , δ as in (*) can even be established on general smooth compact oriented Riemannian manifolds; see the discussion before Theorem 2.4 of [19].) By using Theorem 6.1, one can prove the following result, which generalizes Theorem 2.4 of [19] (when the manifold there is the sphere). Theorem 6.3 (a) Fix a > 1, and say c0 , δ 0 are as in (*) above. Suppose f ∈ S(R+ ), and f (0) = 0. Let Kt be the kernel of f (t2 ∆s ), and let wtxR be as in (117) and (118) above. Then there exists a constant C0 > 0 (depending only on f, a, c0 and δ 0 ) as follows: Let J ⊆ Z be either finite, or equal to all of Z. Say 0 < b < 1. For each j ∈ J , write S 2 as a finite disjoint union of measurable sets {Ej,k : 1 ≤ k ≤ Nj }, where: the diameter of each Ej,k is less than or equal to baj ,

(123)

for each j with baj < δ 0 , µ(Ej,k ) ≥ c0 (baj )2 .

(124)

and where: For each j, k, select xj,k ∈ Ej,k and select any Rj,k ∈ SO(3) with xj,k ∈ URj,k . Set ϕj,k = waj ,xj,k ,Rj,k ∈ C ∞ (Ls ). For F ∈ L2 (Ls ), set

S J F :=

XX

j∈J

k

µ(Ej,k )hF, ϕj,k iϕj,k .

(125)

29 Here, if J P = Z, the series converges unconditionally in L2 (Ls ). J Let Q = j∈J |f |2 (a2j ∆s ); if J = Z, this series converges strongly on L2 (Ls ). Then for all F ∈ L2 (Ls ), h(QJ − S J )F, F i ≤ C0 bhF, F i

(126)

(or, equivalently, since QJ − S J is self-adjoint, kQJ − S J k ≤ C0 b.) (b) In (a), take J = Z; set Q = QZ , S = S Z . Suppose that the Daubechies condition (5) holds. Let P = P|s|,s be the projection onto the H|s|,s , the null space of ∆s (in C ∞ (Ls )). Then (Aa − C0 b)(I − P ) ≤ S ≤ (Ba + C0 b)(I − P )

(127)

as operators on L2 (Ls ). Thus, for any F ∈ (I − P )L2 (M), X (Aa − C0 b)kF k2 ≤ µ(Ej,k )|hF, ϕj,k i|2 ≤ (Ba + C0 b)kF k2 , j,k

so that, if Aa − C0 b > 0, then µ(Ej,k )1/2 ϕj,k j,k is a frame for (I − P )L2 (Ls ), with frame bounds Aa − C0 b and Ba + C0 b. Remarks 1. By (6) and (7), Ba /Aa = 1 + O(|(a − 1)2 (log |a − 1|)|); evidently (Ba + C0 b)/(Aa − C0 b) can be made arbitrarily close to Ba /Aa by choosing b sufficiently small. Thus, Theorem 6.3 gives “nearly tight” frames for (I − P )L2 (Ls ). 2. The proof of Theorem 6.3 follows along the same lines of the proof for s = 0, which is Theorem 2.4 of [19]. It will be given elsewhere [21]. In this article, we will only use Theorem 6.3 in Section 8. 3. In the statement of Theorem 2.4 (a) of [19], J was not assumed to be a finite set, but rather, a cofinite set (that is, J c was assumed to be finite), and one had unconditional convergence of the sum for S J F and strong convergence of the sum for QJ . Note, however, that this is equivalent to (a) for finite sets, c c since QJ = QZ − QJ , S J F = S Z F − S J F .

7

Asymptotic Uncorrelation

We define a random spin s field G(.) by assuming that there exists a probability space (Ω, ℑ, P ) such that the map (x, ω) → G(x, ω) is B(S 2 ) ⊗ ℑ measurable, ℑ denoting a σ-algebra on Ω and B(S 2 ) the Borel σ-field of S 2 . (To clarify: G(x, ω) is a.e. in Lsx (the fiber of Ls above x), and to say that the map is measurable is to say that, for any R ∈ SO(3) the complex-valued function over UR × Ω obtained from the map by trivializing the bundle over UR is measurable.) As is customary, the dependence of G upon ω will be suppressed in the notation. Also, we assume that Z 2 E |G(x)| dS(x) = C < ∞ , S2

which in particular entails that x → G(x) belongs to L2 (Ls ) with probability one (see for instance [40]). In cosmological applications it is natural to introduce an isotropy condition: Definition 7.1 Let G be a random spin s field. We say that G is isotropic (in law) if for every R x1 , . . . , xN ∈ UI , the joint probability distribution of GR I (x1 ), . . . , GI (xN ) is independent of R ∈ SO(3). Similarly, if G, H are both random spin s fields defined on the same probability space, we say that G, H are jointly isotropic (in law) if for every x1 , . . . , xN , y1 , . . . , yM ∈ UI , the joint probability distribution of R R R GR I (x1 ), . . . , GI (xN ), HI (y1 ), . . . , HI (yM ) is independent of R ∈ SO(3). Note that G is isotropic if and only if, for every R′ ∈ SO(3), and for every p1 , . . . , pN ∈ UR′ , the joint probR ′ ability distribution of GR R′ (p1 ), . . . , GR′ (pN ) is independent of R ∈ SO(3). Indeed, say pj = R xj , xj ∈ UI ′ RR (for j = 1, . . . , N ). One need only then observe that, by (28), GR (xj ) for each j. Similarly, R′ (pj ) = GI ′ G, H are jointly isotropic if and only if, for every R ∈ SO(3), and for every p1 , . . . , pN , q1 , . . . , qM ∈ UR′ ,

30 R R R the joint probability distribution of GR R′ (p1 ), . . . , GR′ (pN ), GR′ (q1 ), . . . , GR′ (qN ) is independent of R ∈ SO(3).

If G is isotropic, then for any R′ ∈ SO(3) and any p ∈ UR′ , the probability distribution of GR R′ (p) is independent of R ∈ SO(3). Restricting R to the subgroup of SO(3) which fixes p, we see in particular that GR′ (p) has the same probability distribution as GR′ R (p) = eisψ GR′ (p), where ψ is the angle from ρR′ (p) to ρR′ R (p). If s 6= 0, we may take ψ with eisψ = −1. We then see that the probability distributions of ℜGR′ (p) is the same as that of −ℜGR′ (p), similarly for ℑGR′ (p); in particular, both ℜGR′ (p) and ℑGR′ (p) must have expectation 0. Again if s 6= 0, we may take ψ with eisψ = i, and we see now that ℜGR′ (p) and ℑGR′ (p) have the same probability distributions. Say G is isotropic, and that GR has spin s spherical harmonic coefficients aR lm (for R ∈ SO(3)). Then, R evidently, the aR all have the same probability distribution, and each a has expectation zero. (Here lm lm R R ′ 2 ′ i (for any R ∈ SO(3)) where h i is the inner product on S .). aR = hG , Y i = hG , Y ′ s lm s lmR lm R −1

R If we take R to be a rotation about N, we see (from aR lm = hG, s Ylm i) and from (60) that, if m 6= 0, iθ then the probability distribution of alm is the same as that of e alm for any θ. In particular, ℜalm and ℑalm will have the same probability distributions, if m 6= 0. R R R ′ ′ If G, H are jointly isotropic, aR lm = hG , s Ylm i, blm = hH , s Ylm i, then for any l, m, l , m the covariances R

E(aR lm bl′ m′ ) are evidently independent of R.

In fact, as in the case s = 0 of [6], we can say much more about these covariances: Theorem 7.2 Suppose G, H are jointly isotropic random spin s fields. Let alm = hGR , s Ylm i, and blm = hH R , s Ylm i. Then E(alm bl′ m′ ) = 0 unless l = l′ and m = m′ . Moreover E(alm blm ) does not depend on m; we will denote it by Cl,G,H . Proof We define Wigner’s Dl matrices by R s Ylm

=

X

l Dm ′ m (R) s Ylm′ .

(128)

m

R These matrices surely exist and are unitary, since the s Ylm form an orthonormal basis of Hls whenever R ∈ SO(3) and l ≥ |s|. The matrices are independent of s, as one (128), use of the P seesl at once from R′ RR′ D (R) Y for any R, R′ , we ∂ and ∂ operators, and (67), (68), (40), (55). Since s Ylm = ′ ′ s m m lm m l see that in fact the map R → Dm′ m (R) is a unitary representation of SO(3), see also [45] and [48]. This representation is evidently isomorphic to the action of SO(3) on Hl0 , which as is well known, is irreducible; moreover, the representations for different l are not equivalent. R R R ′ ll′ Now let aR by lm = hG , s Ylm i, and blm = hH , s Ylm i. For each l, l define the covariance matrix M ′

R

−1

R R R ll i, one easily computes that Mmm ′ = E(alm bl′ m′ ) = E(alm bl′ m′ ) for any R. Noting that alm = hG, s Ylm ll′ l ∗ ll′ l M = [D (R)] M D (R) for any R. The theorem now follows at once from Schur’s lemma.

In cosmological applications, having an“asymptotic uncorrelation” theorem of the following kind is of great importance. Generalizing a key result in [4], we now show: Theorem 7.3 Let G, H be random, jointly isotropic spin s fields. If x ∈ UR , let wtxR be the spin wavelet of (117), where f has compact support away from the origin. Assume that Cl,G,H = g(l) for a smooth function g on the interval (|s|, ∞), which satisfies the following condition, for some α > 2: for every i ∈ N0 there exists ci > 0 such that |g (i) (u)| ≤ ci u−α−i . (129) Assume also that, for some c > 0, certain ρ, τ ∈ R, and for all sufficiently large l, Cl,G,G ≥ cl−ρ , and Cl,H,H ≥ cl−τ .

(130)

31 For x ∈ UR , let β G,t,x,R =< G, wtxR >, β H,t,x,R =< H, wtxR >. Then for any M ∈ N0 , there exists T0 > 0 and a constant CM > 0, such that for all x, y ∈ UR , 0 < t < T0 , we have |Cor(β G,t,x,R , β H,t,y,R )| ≤ tα−(ρ+τ )/2

CM . (d(x, y)/t)M

(131)

In particular, for fixed x, y, |Cor(β G,t,x,R , β H,t,y,R )| → 0 as t → 0+ . Remark: Note that |Cor(β t,x,R , β t,y,R )| is independent of R. Proof By definition, |E(β G,t,x,R β H,t,y,R )| q |Cor(β G,t,x,R , β H,t,y,R )| = q . E(|β G,t,y,R |2 ) E(|β H,t,y,R |2 )

Say G, H have spin s spherical harmonic coefficients alm , blm respectively. Note that XX β G,t,x,R = f (t2 λls )alm s YlmR (x),

(132)

(133)

l≥|s| m

and similarly for H. Accordingly X

E(β G,t,x,R β H,t,y,R ) =

l≥|s|

X

=

l≥|s|

|f |2 (t2 λls )Cl,G,H

X

s YlmR (x)s YlmR (y)

(134)

m

ls |f |2 (t2 λls )Cl,G,H KR,R (x, y).

(135)

To estimate E(|β G,t,x,R |2 ), let us note that, since f 6= 0, we may choose c0 and 0 < a < b such that √ 2 2 2 2 a′ , b′ with a < a′ < b′ < b. Since |f |2 ≥ c0 on √ [a , b ]. Thus |f | (t λls ) ≥ c0 if a/t ≤ λls ≤ b/t. Choose 2 2 liml→∞ l/ λls = 1, there is a T1 > 0 such that if 0 < t < T1 , |f | (t λls ) ≥ c0 if a′ /t ≤ l ≤ b′ /t. Choose 0 < T2 < T1 with (b′ − a′ )/t > 1 if 0 < t < T2 . Then, for t sufficiently small, by Theorem 5.5 and (130), for some C ′ , C > 0, E(|β G,t,x,R |2 ) ≥

a′ t

X

c0 l−ρ

′ ≤l≤ bt

b′ − a′ α −1 2l + 1 ≥ C′ t t = Ct−2+ρ . 4π t

(136)

Similarly, E(|β H,t,x,R |2 ) ≥ Ct−2+τ .

(137)

We estimate the numerator in (132) by using Theorem 6.1. In order to do this, we must transform g in some simple ways. Say s ≥ 0. Recall p that λls = (l − s)(l + s + 1) = l(l + 1) − s(s + 1). Thus, if v > 0, v = λls if and only if l = 12 [−1 + 1 + 4[v + s(s + 1)] := Q(v), say. Then Q : (0, ∞) → (0, ∞) bijectively. For v > 0, set h(v) = g(Q(v)); then g(l) = h(λls ). Moreover, it is easy to see, from (129), that for every i ∈ N0 there exists Ci′ > 0 such that di (138) | i h(v)| ≤ Ci′ v −α/2−i dv From (135), we have X ls |f |2 (t2 λls )h(λls )KR,R (x, y) (139) E(β txR β tyR ) = =

l≥|s| α

t Kt,R,R (x, y),

(140)

32 where Kt is the kernel of f[t] (t2 ∆s ), where now f[t] (w) = |f |2 (w)[t−α h(w/t2 )],

(141)

(since |f |2 (t2 λls )h(λls ) = tα f[t] (t2 λls )). Select 0 < A < B with suppf ⊆ [A, B]. Then suppf[t] ⊆ [A, B] for all t. By use of (138), one sees readily that for any L ∈ N0 , there is a C > 0 such that kf[t] kC L ≤ C ′ for all t. Accordingly, by Theorem 6.1 and Remark 6.2, for every M there exists CM > 0 with |E(β txR β tyR )| ≤

′ α −2 CM t t (d(x, y)/t)M

Using this in (132) together with (136), we find the desired result. Similarly for s < 0. Remarks 1. Since suppf[t] ⊆ [A, B], when deriving the estimate kf[t] kC L ≤ C from (138) for t small, one need only use (138) for v large (by (141)). Thus, instead of assuming g is smooth on (|s|, ∞), and that (129) holds there, one may assume only that for some T > 0, g is smooth on (T, ∞), and that (129) holds there. Equivalently, we are assuming that g(u) agrees with an ordinary symbol of order −α for large u. 2. By remark (6.2), an examination of the proof of the theorem shows that for each M there is an L ∈ N0 such that, instead of assuming that g is smooth and satisfies (129) for all i, we need only assume that g ∈ C L and satisfies (129) for all i ≤ L. 3. The hypothesis (129), which is used only to estimate the derivatives of h and then the derivatives of f[t] , can evidently be relaxed – at a cost – to |g (i) (u)| ≤ ci u−αi whenever i ≤ L, for certain αi . The cost in doing this is that one would need to multiply the right side of (131) by a factor of t−N for some sufficiently large N . It would still evidently follow that for fixed x, y, |Cor(β G,t,x,R , β H,t,y,R )| → 0 as t → 0+ . 4. If G = H, in applications of the theorem, it is natural to assume α = ρ = τ . 5. In cosmological applications, where we have (122), and where s = 2, one assumes that the polarization field is (a single sample of) an isotropic random field F , and that FE and FM are jointly isotropic.

8

Stochastic Limit Theorems

In this final section, we give a brief glimpse of the significance of the results of the previous section in cosmology. We prove some stochastic limit results, and then indicate why they are important. Say ǫ > 0. We use a nearly tight frame, as provided by Theorem 6.3.P Specifically, in that theorem, we fix a real ∞ 2 2j f supported in [1/a2 , a2 ] for which the Daubechies sum j=−∞ f (a u) = 1 for all u > 0, so that Aa = Ba = 1. We then choose b sufficiently small that C0 b < ǫ. We then produce {φj,k } as in Theorem 6.3, and let φj,k = µ(Ej,k )1/2 ϕj,k , so that {φj,k } is a nearly tight frame for (I − P )L2 (Ls ), with frame bounds 1 − ǫ and 1 + ǫ. Also, we have (126) for general finite subsets J . Of course our choice of {φj,k } depends on Pǫ; if we want to indicate the dependence on ǫ, we will write φj,k,ǫ . Say G = Alm s Ylm is an isotropic random spin s field, and let Cl = Cl,G,G (notation as in Theorem 7.3). For j ∈ Z, we also let X γj = Cl (2l + 1). (142) a−2 ≤a2j λls ≤a2

(To avoid confusion, the summation in (142), and similar summations in the sequel, are over those l for which λls is defined and for which a−2 ≤ a2j λls ≤ a2 ; recall that λls is defined for l ≥ |s|.) Note that, for any l, there at most 3 values of j for which a−2 ≤ a2j λls ≤ a2 , so that X X Cl (2l + 1) = 3VarG. (143) γj ≤ 3 j

l≥|s|

33 We let β jk = β jkǫ = hG, φj,k,ǫ i. We focus on the quadratic statistics X 2 ˜j = β jk Γ

(144)

k

and

Evidently

We have:

bj ) = E(Γ

X

bj = Γ

X

l≥|s|, m

f 2 (a2j λls )|Alm |2 .

a−2 ≤a2j λ

l≥|s|

bj − Γ ˜ j |) ≤ ǫγ j for all j. Proposition 8.1 E(|Γ

Proof For each j ∈ Z, let

X

f 2 (a2j λls )Cl (2l + 1) =

Gj =

X

(145)

f 2 (a2j λls )Cl (2l + 1). ls

(146)

≤a2

Alm s Ylm .

(147)

a−2 ≤a2j λls ≤a2

Then β jk = hG, φj,k i = µ(Ej,k )1/2

XX

l≥|s| m

f (a2j λls )Alm s Ylm = hGj , φj,k i.

Note also that (I − P )Gj = Gj , since no term with l = |s| can appear in the summation in (147) (for then λls = 0). By Theorem 6.3 (a) with J = {j}, we see that, in the notation of that theorem, ˜ j = hS J Gj , Gj i, Γ so that, by Theorem 6.3 (a),

b j = hQJ Gj , Gj i, Γ bj − Γ ˜ j | ≤ ǫkGj k22 = ǫ |Γ

X


|Alm |2 .

The proposition now follows at once if we take expectations of both sides. ˜ j = P β jkǫ 2 depends on our choice of ǫ, and we can even let ǫ 0 and some α > 2, cl−α ≤ Cl ≤ Cl−α for l > 0. Then bj − E Γ bj Γ r n o →d N (0, 1) as j → −∞ , bj V ar Γ

→d denoting as usual convergence in probability law, and N (0, 1) denoting as usual the standard normal distribution. Proof Write

 √  2f (a2j λls )ℜAlm for m > 0 f (a2j λls )Al0 for m = 0 ; Xlm (j) :=  √ 2f (a2j λls )ℑAlm for m < 0.

For each j, {Xlm (j)} is a triangular array of independent heteroscedastic (= unequal variance) Gaussian random variables such that X 2 2 bj = Xlm (j) , EXlm (j) = f 2 (a2j λls )Cl , Γ l≥|s|, m

V ar

Hence we have easily

2 Xlm (j)

n o bj = V ar Γ =

2 4 2 (j) = 2f 4 (a2j λls )(Cl )2 . = EXlm (j) − EXlm X

l≥|s|, m

X

X 2 2f 4 (a2j λls )(Cl )2 V ar Xlm (j) = l≥|s|, m

2f 4 (a2j λls )(Cl )2 (2l + 1) .

l≥|s|

To establish the Central Limit Theorem, it is then enough to check the validity of the Lindeberg-Levy condition ([10]) which here takes the simple form 2 maxa−2 ≤a2j λls ≤a2 maxm V ar(Xlm ) P 4 2j 2 j→−∞ a−2 ≤a2j λls ≤a2 2f (a λls )(Cl )

lim

≤ ≤ This completes the proof.

maxa−2 ≤a2j λls ≤a2 maxm f 4 (a2j λls )(Cl )2 P 4 2j 2 j→−∞ a−2 ≤a2j λls ≤a2 f (a λls )(Cl ) (2l + 1)

K lim

K lim

j→−∞

a

P 2jα

a2jα


f 4 (a2j λls )l

=0.

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